Adiabatic quantum computation with superconducting qubits

ABSTRACT

A computer program product with computer program mechanism embedded therein is provided. The mechanism has instructions for initializing a quantum system, which includes a plurality of qubits, to an initialization Hamiltonian H O . The system is capable of being in one of at least two configurations at any give time including H O  and a problem Hamiltonian H P . Each respective first qubit in the plurality of qubits is arranged with respect to a respective second qubit in the plurality of qubits such that the first respective qubit and the second respective qubit define a predetermined coupling strength. The predetermined coupling strengths between the qubits in the plurality of qubit collectively define a computational problem to be solved. The mechanism further comprises instructions for adiabatically changing the system until it is described by the ground state of the problem Hamiltonian H P  and instructions for reading out the state of the system.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims benefit, under 35 U.S.C. § 119(e), of U.S.Provisional Patent Application No. 60/557,748, filed on Mar. 29, 2004,which is hereby incorporated by reference in its entirety. Thisapplication also claims benefit, under 35 U.S.C. § 119(e), of U.S.Provisional Patent Application No. 60/588,002, filed on Jul. 13, 2004,which is hereby incorporated by reference in its entirety. Thisapplication is further related to concurrently filed application Ser.No. ______, Attorney Docket No. 706700-999193, entitled “AdiabaticQuantum Computation with Superconducting Qubits,” and application Ser.No. ______, Attorney Docket No. 706700-999207, entitled “AdiabaticQuantum Computation with Superconducting Qubits,” each of which ishereby incorporated by reference in its entirety.

FIELD OF THE INVENTION

This invention relates to superconducting circuitry. More specifically,this invention relates to devices for quantum computation.

BACKGROUND

Research on what is now called quantum computing may have begun with apaper published by Richard Feynman. See Feynman, 1982, InternationalJournal of Theoretical Physics 21, pp. 467-488, which is herebyincorporated by reference in its entirety. Feynman noted that a quantumsystem is inherently difficult to simulate with conventional computersbut that observation of the evolution of an analogous quantum systemcould provide an exponentially faster way to solve the mathematicalmodel of the quantum system of interest. In particular, solving amathematical model for the behavior of a quantum system commonlyinvolves solving a differential equation related to the Hamiltonian ofthe quantum system. David Deutsch noted that a quantum system could beused to yield a time savings, later shown to include exponential timesavings, in certain computations. If one had a problem modeled in theform of an equation that represented the Hamiltonian of a quantumsystem, the behavior of the system could provide information regardingthe solutions to the equation. See Deutsch, 1985, Proceedings of theRoyal Society of London A 400, pp. 97-117, which is hereby incorporatedby reference in its entirety.

A major activity in the quantum computing art is the identification ofphysical systems that can support quantum computation. This activityincludes finding suitable qubits as well as developing systems andmethods for controlling such qubits. As detailed in the followingsections, a qubit serves as the basis for performing quantumcomputation.

2.1 Qubits

The physical systems that are used in quantum computing are quantumcomputers. A quantum bit or “qubit” is the building block of a quantumcomputer in the same way that a conventional binary bit is a buildingblock of a classical computer. A qubit is a quantum bit, the counterpartin quantum computing to the binary digit or bit of classical computing.Just as a bit is the basic unit of information in a classical computer,a qubit is the basic unit of information in a quantum computer. A qubitis conventionally a system having two or more discrete energy states.The energy states of a qubit are generally referred to as the basisstates of the qubit. The basis states of a qubit are termed the |0> and|1> basis states. In the mathematical modeling of these basis states,each state is associated with an eigenstate of the sigma-z (σ^(Z)) Paulimatrix. See Nielsen and Chuang, 2000, Quantum Computation and QuantumInformation, Cambridge University Press, which is hereby incorporated byreference in its entirety.

The state of a qubit can be in any superposition of two basis states,making it fundamentally different from a bit in an ordinary digitalcomputer. A superposition of basis states arises in a qubit when thereis a non-zero probability that the system occupies more than one of thebasis states at a given time. Qualitatively, a superposition of basisstates means that the qubit can be in both basis states |0> and |1> atthe same time. Mathematically, a superposition of basis states meansthat the wave function that characterizes the overall state of thequbit, denoted |Ψ>, has the form|Ψ>=a|0>+b|1>where a and b are amplitudes respectively corresponding to probabilities|a|² and |b|². The amplitudes a and b each have real and imaginarycomponents, which allows the phase of qubit to be modeled. The quantumnature of a qubit is largely derived from its ability to exist in asuperposition of basis states, and for the state of the qubit to have aphase.

To complete a quantum computation using a qubit, the state of the qubitis typically measured (e.g., read out). When the state of the qubit ismeasured the quantum nature of the qubit is temporarily lost and thesuperposition of basis states collapses to either the |0> basis state orthe |1> basis state, thus regaining its similarity to a conventionalbit. The actual state of the qubit after it has collapsed depends on theamplitudes a and b immediately prior to the readout operation.

A survey of exemplary physical systems from which qubits can be formedis found in Braunstein and Lo (eds.), Scalable Quantum Computers,Wiley-VCH Verlag GmbH, Berlin (2001), which is hereby incorporated byreference in its entirety. Of the various physical systems surveyed, thesystems that appear to be most suited for scaling (e.g., combined insuch a manner such that they entangle with each other) are thosephysical systems that include superconducting structures such assuperconducting qubits.

2.2 Superconducting Qubits in General

Superconducting qubits generally fall into two categories; phase qubitsand charge qubits. Phase qubits store and manipulate information in thephase states of the device. Charge qubits store and manipulateinformation in the elementary charge states of the device. Insuperconducting materials, phase is a property of the material whereaselementary charges are represented by pairs of electrons called Cooperpairs. The division of such devices into two classes is outlined inMakhlin et al., 2001, “Quantum-State Engineering with Josephson-JunctionDevices,” Reviews of Modern Physics 73, pp. 357-401 which is herebyincorporated by reference in its entirety.

Phase and charge are related values in superconductors and, at energyscales where quantum effects dominate, the Heisenberg uncertaintyprinciple causes certainty in phase to lead to uncertainty in chargeand, conversely, causes certainty in charge to lead to uncertainty inthe phase of the system. Superconducting phase qubits are devices formedout of superconducting materials having a small number of distinct phasestates and many charge states, such that when the charge of the deviceis certain, information stored in the phase states becomes delocalizedand evolves quantum mechanically. Therefore, fixing the charge of aphase qubit leads to delocalization of the phase states of the qubit andsubsequent useful quantum behavior in accordance with well-knownprinciples of quantum mechanics.

Experimental realization of superconducting devices as qubits was madeby Nakamura et al., 1999, Nature 398, p. 786, which is herebyincorporated by reference in its entirety. Nakamura et al. developed acharge qubit that demonstrates the basic operational requirements for aqubit. However, the Nakamura et al. charge qubits have unsatisfactorilyshort decoherence times and stringent control parameters. Decoherencetime is the duration of time that it takes for a qubit to lose some ofits quantum mechanical properties, e.g., the state of the qubit nolonger has a definite phase. When the qubit loses it quantum mechanicalproperties, the phase of the qubit is no longer characterized by asuperposition of basis states and the qubit is no longer capable ofsupporting all types of quantum computation.

Superconducting qubits have two modes of operation related tolocalization of the states in which information is stored. When thequbit is initialized or measured, the information is classical, 0 or 1,and the states representing that classical information are alsoclassical in order to provide reliable state preparation. Thus, a firstmode of operation of a qubit is to permit state preparation andmeasurement of classical information. A second mode of operation occursduring quantum computation, where the information states of the devicebecome dominated by quantum effects such that the qubit can evolvecontrollably as a coherent superposition of those states and, in someinstances, even become entangled with other qubits in the quantumcomputer. Thus, qubit devices provide a mechanism to localize theinformation states for initialization and readout operations, andde-localize the information states during computation. Efficientfunctionality of both of these modes and, in particular, the transitionbetween them in superconducting qubits is a challenge that has not beensatisfactorily resolved in the prior art.

2.2.1 Phase Qubits

A proposal to build a quantum computer from superconducting qubits waspublished in 1997. See Bocko et al., 1997, IEEE Trans. Appl. Supercon.7, p. 3638, which is hereby incorporated by reference in its entirety.See also Makhlin et al., 2001, Rev. Mod. Phys. 73, p. 357 which ishereby incorporated by reference in its entirety. Since then, designsbased on many other types of qubits have been introduced. One suchdesign is based on the use of superconducting phase qubits. See Mooij etal., 1999, Science 285, 1036; and Orlando et al., 1999, Phys. Rev. B 60,15398, which are hereby incorporated by reference in their entireties.In particular, quantum computers based on persistent current qubits,which are one type of superconducting phase qubit, have been proposed.

The superconducting phase qubit is well known and has demonstrated longcoherence times. See, for example, Orlando et al., 1999, Phys. Rev. B60, 15398, and Il'ichev et al., 2003, Phys. Rev. Lett. 91, 097906, whichare hereby incorporated by reference in their entireties. Some othertypes of superconducting phase qubits comprise superconducting loopsinterrupted by more or less than three Josephson junctions. See, e.g.,Blatter et al., 2001, Phys. Rev. B 63, 174511, and Friedman et al.,2000, Nature, 406, 43, which are hereby incorporated by reference intheir entireties.

FIG. 1A illustrates a persistent current qubit 101. Persistent currentqubit 101 comprises a loop 103 of superconducting material interruptedby Josephson junctions 101-1, 101-2, and 101-3. Josephson junctions aretypically formed using standard fabrication processes, generallyinvolving material deposition and lithography stages. See, e.g., Madou,Fundamentals of Microfabrication, Second Edition, CRC Press, 2002, whichis hereby incorporated by reference in its entirety. Methods forfabricating Josephson junctions are well known and described in Ramos etal., 2001, IEEE Trans. App. Supercond. 11, 998, for example, which ishereby incorporated by reference in its entirety. Details specific topersistent current qubits can be found in C. H. van der Wal, 2001; J. B.Majer, 2002; and J. R. Butcher, 2002, all Theses in Faculty of AppliedSciences, Delft University of Technology, Delft, The Netherlands;http://qt.tn.tudelft.nl; Kavli Institute of Nanoscience Delft, DelftUniversity of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands,which is hereby incorporated by reference in its entirety. Commonsubstrates include silicon, silicon oxide, or sapphire, for example.Josephson junctions can also include insulating materials such asaluminum oxide, for example. Exemplary superconducting materials usefulfor forming superconducting loop 103 are aluminum and niobium. TheJosephson junctions have cross-sectional sizes ranging from about 10nanometers (nm) to about 10 micrometers (μm). One or more of theJosephson junctions 101 has parameters, such as the size of thejunction, the junction surface area, the Josephson energy or thecharging energy that differ from the other Josephson junctions in thequbit.

The difference between any two Josephson junctions in the persistentcurrent qubit is characterized by a coefficient, termed α, whichtypically ranges from about 0.5 to about 1.3. In some instances, theterm α for a pair of Josephson junctions in the persistent current qubitis the ratio of the critical current between the two Josephson junctionsin the pair. The critical current of a Josephson junction is the minimumcurrent through the junction at which the junction is no longersuperconducting. That is, below the critical current, the junction issuperconducting whereas above the critical current, the junction is notsuperconducting. Thus, for example, the term α for junctions 101-1 and101-2 is defined as the ratio between the critical current of junction101-1 and the critical current of junction 101-2.

Referring to FIG. 1A, a bias source 110 is inductively coupled topersistent current qubit 101. Bias source 110 is used to thread amagnetic flux Φ_(x) through phase qubit 101 to provide control of thestate of the phase qubit. In some instances, the persistent currentqubit operates with a magnetic flux bias Φ_(x) ranging from about 0.2·Φ₀to about 0.8·Φ₀, where Φ₀ is the flux quantum. In some instances, themagnetic flux bias ranges from about 0.47·Φ₀ to about 0.5·Φ₀.

Persistent current qubit 101 has a two-dimensional potential withrespect to the phase across Josephson junctions 101-1, 101-2, and 101-3.In some instances, persistent current qubit 101 is biased with amagnetic flux Φ_(x), such that the two-dimensional potential profileincludes regions of local energy minima, where the local energy minimaare separated from each other by small energy barriers and are separatedfrom other regions by large energy barriers. In some instances, thispotential has the shape of double well potential 100B (FIG. 1B), whichincludes a left well 160-0 and a right well 160-1. In such instances,left well 160-0 can represent clockwise (102-0) circulating supercurrentin the phase qubit 101 and right well 160-1 can representcounter-clockwise (102-1) circulating supercurrent in persistent currentqubit 101 of FIG. 1A.

When wells 160-0 and 160-1 are at or near degeneracy, meaning that theyare at the same or nearly the same energy potential as illustrated inFIG. 1B, the quantum state of persistent current qubit 101 becomes acoherent superposition of the phase or basis states and device can beoperated as a phase qubit. The point at or near degeneracy is hereinreferred to as the point of computational operation of the persistentcurrent. During computational operation of the persistent current qubit,the charge of the qubit is fixed leading to uncertainty in the phasebasis and delocalization of the phase states of the qubit. Controllablequantum effects can then be used to process the information stored inthose phase states according to the rules of quantum mechanics. Thismakes the persistent current qubit robust against charge noise andthereby prolongs the time under which the qubit can be maintained in acoherent superposition of basis states.

2.2.2 Charge Qubits

There are broad classes of condensed matter systems that have statesdefined by the presence and absence of extra charge, or the excesscharge exists in either a ground or an excited state. Such systems arediverse and have long held theoretical and experimental interest, e.g.Millikan, 1911, Phys. Rev. 32, pp. 349-397, which is hereby incorporatedby reference in its entirety. There has been attention directed tosemiconductor systems such as quantum dots. Research has been conductedusing single particle electronics because they hold promise forconventional computers. Subsequently, proposals were made for thesesystems as quantum computers. While these systems collectively could becalled charge qubits, this term is reserved herein for superconductingqubits. Specifically, a superconducting charge qubit has as basis statesthe presence (charge=2e, or some multiple thereof of 2e) or absence(charge=0) of charge on a small superconducting island. For chargequbits, the Coulomb energy E_(C)=e²/2C exceeds the Josephson energyE_(J), of the qubit.

A charge qubit is a small (mesoscopic) island of superconductorseparated by a Josephson junction from a large superconductor(reservoir), see FIG. 12, for example. This system can be tuned tobehave like an ideal two-level quantum system. Classical basis states|0> and |1> (corresponding to the presence or absence of a Cooper pair)are the working states of the charge qubit. The Hamiltonian of thesuperconducting charge qubit is,$H = {- {\frac{1}{2}\left\lbrack {{4{E_{C}\left( {1 - {2n_{g}}} \right)}\quad\sigma^{Z}} + {E_{J}\sigma^{X}}} \right\rbrack}}$where the dimensionless gate charge n_(g)≈C_(g)V_(g)/2e is determined inoperation by the gate voltage V_(g), and in fabrication by thecapacitance C_(g). Here it is assumed that n_(g)≈½. The bias term forthe charge qubit is proportional to σ^(Z). A finite Josephson energyallows transition between the states with tunnel splitting Δproportional to E_(J). The Josephson energy of the Josephson junctionconnecting the island to the reservoir can be made tunable. Tuning thedimensionless gate and the Josephson energy allows one to independentlybias the charge and vary the tunneling rate of the charge qubit. SeeNakamura et al., 1999, Nature 398, pp. 786-788; and Makhlin et al.,2001, Rev. Mod. Phys. 73, pp. 357-401, each of which is herebyincorporated by reference in its entirety.

2.3 NP Complexity Classes

Computer scientists concerned with complexity routinely use thedefinitions of different complexity classes. The number of complexityclasses is ever changing, as new ones are defined and existing onesmerge through advancements made in computer science. The complexityclasses known as non-deterministic polynomial-time (NP), NP-complete(NPC), and NP-hard (NPH) are all classes of decision problems. Decisionproblems have binary outcomes.

Problems in NP are computational problems for which there existspolynomial time verification. That is, it takes no more than polynomialtime (class P) in the size of the problem to verify a potentialsolution. It may take more than polynomial time to create a potentialsolution. NP-hard problems take longer to verify a potential solution.For each NP-hard problem, there is an NP-complete problem that can bereduced to the NP-hard problem. However, NP-complete problems that canbe reduced to a NP-hard problem do not enjoy polynomial timeverification.

Problems in NPC can be defined as problems in NP that have been shown tobe equivalent to, or harder to solve, than a known problem in NPC.Equivalently, the problems in NPC are problems in NP that are also inNPH. This can be expressed as NPC=NP ∩ NPH.

A problem is equivalent, or harder to solve, than a known problem in NPCif there exists a polynomial time reduction to the instant problem fromthe known problem in NPC. Reduction can be regarded as a generalizationof mapping. The mappings can be one to one functions, many to onefunctions, or make use of oracles, etc. The concepts of complexityclasses and how they define the intractability of certain computationalproblems is found in, for example, Garey and Johnson, 1979, Computersand Intractability: A Guide to the Theory of NP-Completeness, Freeman,San Francisco, ISBN: 0716710455, which is hereby incorporated byreference in its entirety. Also see, Cormen, Leiserson, and Rivest,1990, Introduction to Algorithms, MIT Press, Cambridge, ISBN:0262530910.

2.4 Circuit Model of Quantum Computing

Analogous to the way a classical computer is built using wires and logicgates, a quantum computer can be built using quantum circuits comprisedof “wires” and “unitary gates.” Here, the wire is not a physical entity.Rather, it represents the state of the qubit in time. The “unitarygates” are applied at precise times to specific qubits to effectevolution of the qubit in accordance with the circuit model for quantumcomputing. The circuit model of quantum computing is a standard anduniversal model used by many practitioners in the art. The circuit modelis universal in the sense that it is able to convert any input stateinto any output state. The elements of the circuit model are that asmall set of one- and two-qubit unitary gates are applied to the qubitswith precise timing. The circuit model of quantum computing canimplement algorithms such as Shor's algorithm for factoring numbers orGrover's algorithm for searching databases. Shor's algorithm provides anexponential speedup relative to classical (non-quantum) computers forfactoring numbers. Grover's application provides a polynomial speed uprelative to classical computers for searching databases. See, forexample, Nielsen and Chuang, 2000, Quantum Computation and QuantumInformation, Cambridge University Press, which is hereby incorporated byreference in its entirety.

An example of the circuit model is shown in FIG. 2. Circuit 200 is animplementation of the quantum Fourier transform. The quantum Fouriertransform is a useful procedure found in many quantum computingapplications based on the circuit model. See, for example, U.S. patentPublication 2003/0164490 A1, entitled “Optimization process for quantumcomputing process,” which is hereby incorporated by reference in itsentirety. Time progresses from left to right, i.e., time step 201precedes time step 202, and so forth. The four qubits in the quantumsystem described by FIG. 2 are indexed 0-3 from bottom to top. The stateof qubit 0 at any given time step is represented by wire S0-S0′, thestate of qubit 1 at any give time step is represented by S1-S1′, etc. Intime step 201, a single-qubit unitary gate, A₃, is applied to qubit 3.The next gate on wire S3-S3′ for qubit 3 is a two-qubit gate, B₂₃, whichis applied to qubits 2 and 3 at time step 202. In general the A_(i) gate(e.g., A₃ as applied to qubit 3 at time step 201) is a HADAMARD gateapplied on the i^(th) qubit while the B_(ij) gate (e.g., B₂₃ which isapplied to qubits 2 and 3 at time step 202) is a CPHASE gate couplingthe i^(th) and j^(th) qubit. The application of unitary gates continuesuntil states S0-S3 have been converted to S0′-S3′. After time step 210,more unitary gates can be applied to the qubits or the states of thequbits can be determined (e.g., by measurement).

2.5 Adiabatic Model of Quantum Computation

The following subsections discuss the adiabatic theorem of quantummechanics and introduce adiabatic quantum computing.

2.5.1 Adiabatic Theorem of Quantum Mechanics

One definition of an adiabatic process is a process that occurs in asystem without heat entering or leaving the system. There exists atheorem in quantum mechanics that provides a suitable framework for suchprocesses. The adiabatic theorem of quantum mechanics has severalversions but a notable element of many such versions is as follows. Aquantum system prepared in its ground state will remain in the groundstate of the various instantaneous Hamiltonians through which it passes,provided the changes are made sufficiently slowly. This form of changeis termed adiabatic change. Such a system is adiabatic because thepopulation of the various states of the quantum system has not beenaltered as a result of the change. Hence, if the populations have notchanged, the temperature of the system has not changed, and therefore noheat has entered or left the system.

2.5.2 Adiabatic Quantum Computing

In 2000, a form of quantum computing, termed adiabatic quantumcomputing, was proposed. See, for example, Farhi et al., 2001, Science292, pp. 472-475, which is hereby incorporated by reference in itsentirety. In adiabatic quantum computing (AQC), the problem to be solvedis encoded into a physical system such that departures from the solutionto the problem incur a net energy cost to the system. AQC is universalin that it is able to convert any input state into any output state.However, unlike the circuit model of quantum computing, there is noapplication of a predetermined set of one- and two-qubit unitary gatesat precise times. It is believed that AQC can be used to find solutionsto some problems with greater efficiency than the circuit model. Suchproblems include problems contained in, and related to, the NP, NP-hard,and NP-complete classes.

As shown in FIG. 3, AQC involves initializing a system, which encodes aproblem to be solved, to an initial state. This initial state isdescribed by an initial Hamiltonian H₀. Then the system is migratedadiabatically to a final state described by Hamiltonian H_(P). The finalstate encodes a solution to the problem. The migration from H₀ to H_(P)follows an interpolation path described by function γ(t) that iscontinuous over the time interval zero to T, inclusive, and has acondition that the magnitude of initial Hamiltonian H₀ is reduced tozero after time T. Here, T, refers to the time point at which the systemreaches the state represented by the Hamiltonian H_(P). Optionally, theinterpolation can traverse an extra Hamiltonian H_(E) that can containtunneling terms for some or all of the qubits represented by H₀. Themagnitude of extra Hamiltonian H_(E) is described by a function δ(t)that is continuous over the time interval zero to T, inclusive, and iszero at the start (t=0) and end (t=T) of the interpolation while beingnon-zero at all or a portion of the times between t=0 and t=T.

One computational problem that can be solved with adiabatic quantumcomputing is the MAXCUT problem. Consider an undirected edge-weightedgraph having a set of vertices and a set of edges. All the edges in thegraph have weights given by a positive integer. The MAXCUT problem,expressed as a decision problem, asks whether there is a partition ofthe graph such that the sum of the weights of the edges crossing thepartition is equal or greater than some given predefined positiveinteger K. Many other permutations of the problem exist and includeoptimization problems based on this decision problem. An example of anoptimization problem is the identification of the partition of the graphthat yields the maximum K In other words, for graph G=(V, E) that is a(not necessarily simple) undirected edge-weighted graph with nonnegativeweights, where a cut C of G is any nontrivial subset of V, the weight ofcut C is the sum of weights of edges crossing the cut. The MAXCUTproblem, expressed as an optimization problem, is the identification ofa cut G having the maximum possible weight.

The MAXCUT problem, expressed as a decision problem, is defined in Gareyand Johnson, 1979, Computers and Intractability: A Guide to the Theoryof NP-Completeness, Freeman, San Francisco, ISBN: 0716710455, which ishereby incorporated by reference in its entirety, as:

-   -   INSTANCE: Graph G=(V, E), weight w(e) ε Z⁺ for each e ε E, for        positive integer K    -   QUESTION: Is there a partition of V into disjoint sets V₁ and V₂        such that the sum of the weights of the edges form E that have        one endpoint in V₁ and one endpoint in V₂ is at least K?

Consider an instance of a positive number K and a graph G=(V, E), havinga set of vertices V={v₁, . . . , v_(|V|)}, and a set of edges E={e₁, . .. , e_(i), . . . , e_(|E|)}, where e_(i)=(v_(j),v_(k)) for all1<j,k<|V|. The graph's edges have weights w(e_(i)), w(v_(j),v_(k)), orw_(jk) that are positive. The explicit decision problem is whether thereis a partition of V, i.e., V₁ ⊂ V, V₂ ⊂ V, and V₁ ∪ V₂=V, such that thesum of the weights of the edges crossing the partition is equal orgreater than some given predefined positive integer K, e.g.,${\sum\limits_{\forall{v_{j} \in V_{1}}}{\sum\limits_{\forall{v_{k} \in V_{2}}}{w\left( {v_{j},v_{k}} \right)}}} \geq {K.}$An optional addition to the definitions above is the graph may havevertex weights that are also positive w(v_(i)) or w_(i). Using thisalternative, MAXCUT can be formulated as a search for a partition of Gsuch that the sum of the weights of the edges crossing the partition,and the sum of the weight of vertices on one side of the partition isequal or greater than K. MAXCUT is a problem that has been solved usinga nuclear magnetic resonance (NMR) quantum computer. See, for example,M. Steffen, Wim van Dam, T. Hogg, G. Breyta, and I. Chuang, 2003,“Experimental Implementation of an Adiabatic Quantum OptimizationAlgorithm,” Phys. Rev. Lett. 90, 067903, which is hereby incorporated byreference in its entirety.

Mathematically, solving MAXCUT permits optimizations based on MAXCUT tobe solved efficiently. In other words, efficiency in solving adecision-based MAXCUT problem (e.g., is there a cut having a valuegreater than some predetermined given value K) will lead to efficiencyin solving the corresponding optimization-based MAXCUT problem (findingthe cut having the greatest value). This is generally true of anyproblem in NP. However, for problems in NPH, their related optimizationproblems represent a class for which adiabatic quantum computing can beparticularly well suited.

One computational problem that can be solved with adiabatic quantumcomputing is the INDEPENDENT SET problem. Garey and Johnston, 1979,Computers and Intractability: A Guide to the Theory of NP-Completeness,define the INDEPENDENT SET problem as:

-   -   INSTANCE: Graph G=(V, E), positive integer K≦|V|.    -   QUESTION: Does G contain an independent set of size K or more,        i.e., as subset of V′⊂ V with |V′|≧K such that no two vertices        in V′ are joined by an edge in E?        where emphasis is added to show differences between the        INDEPENDENT SET problem and another problem, known as CLIQUE,        that is described below. Expanding upon this definition,        consider an undirected edge-weighted graph having a set of        vertices and a set of edges, and a positive integer K that is        less than or equal to the number of vertices of the graph. The        INDEPENDENT SET problem, expressed as a decision problem, asks        whether there is a subset of vertices of size K, such that no        two vertices in the subset are connected by an edge of the        graph. Many other permutations of the problem exist and include        optimization problems based on this decision problem. An example        of an optimization problem is the identification of the        independent set of the graph that yields the maximum K. This is        called MAX INDEPENDENT SET.

Mathematically, solving INDEPENDENT SET permits optimizations based onINDEPENDENT SET, such as MAX INDEPENDENT SET to be solved efficiently.In other words, efficiency in solving a decision-based INDEPENDENT SETproblem (e.g., is there an independent set having a value greater thatsome predetermined given value K) will lead to efficiency in solving thecorresponding optimization-based INDEPENDENT SET problem (finding theindependent set having the greatest value). This is generally true ofany problem in NP.

Mathematically, solving INDEPENDENT SET permits the solving of yetanother problem known as CLIQUE. This problem seeks the clique in agraph. A clique is a set of vertices that are all connected to eachother. Given a graph, and a positive integer K, the question that isasked in CLIQUE is whether there are K vertices all of which areneighbors of each other. Like the INDEPENDENT SET problem, the CLIQUEproblem can be converted to an optimization problem. The computation ofcliques has roles in economics and cryptography. Solving an independentset on graph G₁=(V, E) is equivalent to solving clique on G₁'scomplement G₂=(V,(V×V)|E), e.g., for all vertices connected by edges inE remove the edges, insert into G₂ edges connecting vertices notconnected in G₁. Garey and Johnston define CLIQUE as:

-   -   INSTANCE: Graph G=(V, E), positive integer K≦|V|.    -   QUESTION: Does G contain a clique of size K or more, i.e., as        subset of V′ ⊂ V with |V′|≧K such that every two vertices in V′        are joined by an edge in E?        Here, emphasis has been added to show differences between CLIQUE        and INDEPENDENT SET. It can also be shown how CLIQUE is related        to the problem VERTEX COVER. Again, all problems in NP-complete        are reducible to each other within polynomial time, making        devices that solve one NP-complete problem efficiently, useful        for other NP-complete problems.

2.6 Adiabatic Quantum Computing Using Superconducting Qubits

The question of whether superconducting qubits can be used to implementadiabatic quantum computing (AQC) has been posed in the art. However,such proposals are unsatisfactory because they either lack enablingdetails on the physical systems on which AQC would be implemented orthey rely on qubits that have not been shown to successfully perform ann-qubit quantum computation, where n is greater than 1 and the quantumcomputation requires entanglement of qubits. For example, Kaminsky andLloyd, 2002, “Scalable Architecture for Adiabatic Quantum Computing ofNP-Hard Problems,” in Quantum Computing & Quantum Bits in MesoscopicSystems, Kluwer Academic, Dordrecht, Netherlands, also published asarXiv.org: quant-ph/0211152, which is hereby incorporated by referencein its entirety, suggests that AQC can be performed with persistentcurrent qubits, without explicitly stating how. As another example, W.M. Kaminsky, S. Lloyd, T. P. Orlando, 2004, “Scalable SuperconductingArchitecture for Adiabatic Quantum Computation,” arXiv.org:quant-ph/0403090, hereby incorporated by reference, describes a methodand structure for AQC for a type of persistent current qubit that hasnot been shown to support multi-qubit quantum computation. Thisreference shows a system of logical qubits without giving explicitconstruction of the logical qubits from physical qubits or an explicitcoupling between more than two qubits.

Accordingly, given the above background, there is a need in the art forimproved systems and methods for adiabatic quantum computing. Discussionor citation of a reference herein shall not be construed as an admissionthat such reference is prior art to the present invention.

SUMMARY OF THE INVENTION

The present invention addresses the need in the art for improved systemsand methods for adiabatic quantum computing. In some embodiments of thepresent invention, a graph based computing problem, such as MAXCUT, isrepresented by an undirected edge-weighted graph. Each node in theedge-weighted graph corresponds to a qubit in a plurality of qubits. Theedge weights of the graph are represented in the plurality of qubits bythe values of the coupling energies between the qubits. For example, theedge weight between a first and second node in the graph is representedby the coupling energy between a corresponding first and second qubit inthe plurality of qubits.

In one aspect of the present invention, the plurality of qubits thatrepresents the graph is initialized to a first state that does notpermit the qubits to quantum tunnel. Then, the plurality of qubits isset to an intermediate state in which quantum tunneling betweenindividual basis states within each qubit in the plurality of qubits canoccur. In preferred embodiments, the change to the intermediate stateoccurs adiabatically. In other words, for any given instant t thatoccurs during the change to the intermediate state or while the qubitsare in the intermediate state, the plurality of qubits are in the groundstate of an instantaneous Hamiltonian that describes the plurality ofqubits at the instant t. The qubits remain in the intermediate statethat permits quantum tunneling between basis states for a period of timethat is sufficiently long enough to allow the plurality of qubits toreach a solution for the computation problem represented by theplurality of qubits.

Once the qubits have been permitted to quantum tunnel for a sufficientperiod of time, the state of the qubits is adjusted such that they reachsome final state that either does not permit quantum tunneling or, atleast, does not permit rapid quantum tunneling. In preferredembodiments, the change to the final state occurs adiabatically. Inother words, for any given instant t that occurs during the change tothe final state, the plurality of qubits are in the ground state of aninstantaneous Hamiltonian that describes the plurality of qubits at theinstant t.

In other examples of the systems and methods of the present invention,the plurality of qubits that represents the graph is initialized to afirst state that does permit the qubits to quantum tunnel. The state ofthe quantum system is changed once the qubits have been permitted toquantum tunnel for a sufficient period of time. The state of the qubitsis adjusted such that they reach some final state that either does notpermit quantum tunneling or, at least, does not permit rapid quantumtunneling. In preferred embodiments, the change to the final stateoccurs adiabatically.

Some embodiments of the present invention are universal quantumcomputers in the adiabatic quantum computing model. Some embodiments ofthe present invention include qubits with single-qubit Hamiltonian termsand at least one two-qubit Hamiltonian term.

A first aspect of the invention provides a method for quantum computingusing a quantum system comprising a plurality of superconducting qubits.The quantum system is characterized by an impedance. Also, the quantumsystem is capable of being in any one of at least two configurations atany given time. These at least two configurations include a firstconfiguration characterized by an initialization Hamiltonian H_(O) aswell as a second configuration characterized by a problem HamiltonianH_(P.) The problem Hamiltonian has a ground state. Each respective firstsuperconducting qubit in the plurality of superconducting qubits isarranged with respect to a respective second superconducting qubit inthe plurality of superconducting qubits such that the first respectivesuperconducting qubit and the corresponding second respectivesuperconducting qubit define a predetermined coupling strength. Thepredetermined coupling strengths between each of the first respectivesuperconducting qubit and corresponding second respectivesuperconducting qubit collectively define a computational problem to besolved. In this first aspect of the invention, the method comprisesinitializing the quantum system to the initialization Hamiltonian H_(O).The quantum system is then adiabatically changed until it is describedby the ground state of the problem Hamiltonian H_(P). The state of thequantum system is then read out by probing an observable of the σ_(X)Pauli matrix operator.

In some embodiments in accordance with the first aspect of theinvention, the reading step comprises measuring an impedance of thequantum system. In some embodiments the reading step comprisesdetermining a state of a superconducting qubit in the plurality ofsuperconducting qubits. In some embodiments, the reading stepdifferentiates a ground state of the superconducting qubit from anexcited state of the superconducting qubit. In some embodiments, asuperconducting qubit in the plurality of superconducting qubits is apersistent current qubit. In some embodiments, the reading step measuresa quantum state of the superconducting qubit as a presence or an absenceof a voltage. In some embodiments, a superconducting qubit in theplurality of superconducting qubits is capable of tunneling between afirst stable state and a second stable state when the quantum system isin the first configuration.

In some embodiments, a superconducting qubit in the plurality ofsuperconducting qubits is capable of tunneling between a first stablestate and a second stable state during the adiabatic changing step. Insome embodiments, the adiabatic changing step occurs during a timeperiod that is between 1 nanosecond and 100 microseconds. In someembodiments, the initializing step includes applying a magnetic field tothe plurality of superconducting qubits in the direction of a vectorthat is perpendicular to a plane defined by the plurality ofsuperconducting qubits. In some embodiments, a superconducting qubit inthe plurality of superconducting qubits is a persistent current qubit.

A second aspect of the invention provides a method for quantum computingusing a quantum system that comprises a plurality of superconductingqubits. The quantum system is coupled to an impedance readout device.The quantum system is capable of being in any one of at least twoconfigurations at any given time. The at least two configurationsinclude a first configuration characterized by an initializationHamiltonian H₀, and a second Hamiltonian characterized by a problemHamiltonian H_(P). The problem Hamiltonian H_(P) has a ground state.Each respective first superconducting qubit in the plurality ofsuperconducting qubits is arranged with respect to a respective secondsuperconducting qubit in the plurality of superconducting qubits suchthat the first respective superconducting qubit and the secondrespective superconducting qubit define a predetermined couplingstrength. The predetermined coupling strength between each said firstrespective superconducting qubit and corresponding second respectivesuperconducting qubit collectively define a computational problem to besolved. In this second aspect of the invention, method comprisesinitializing the quantum system to the initialization Hamiltonian H_(O).Then the quantum system is adiabatically changed until it is describedby the ground state of the problem Hamiltonian H_(P). The state of thequantum system is then read out through the impedance readout devicethereby solving the computational problem.

In some embodiments in accordance with this second aspect of theinvention, the reading step measures a quantum state of asuperconducting qubit in the plurality of superconducting qubits as apresence or an absence of a voltage. In some embodiments, the readingstep differentiates a ground state of the superconducting qubit from anexcited state of the superconducting qubit. In some embodiments, asuperconducting qubit in the plurality of superconducting qubits is (i)a phase qubit in the charge regime or (ii) a persistent current qubit.In some embodiments, a superconducting qubit in the plurality ofsuperconducting qubits is capable of tunneling between a first stablestate and a second stable state when the quantum system is in the firstconfiguration. In some embodiments, a superconducting qubit in theplurality of superconducting qubits is capable of tunneling between afirst stable state and a second stable state during the adiabaticchanging step. In some embodiments, the adiabatic changing step occursduring a time period that is greater than 1 nanosecond and less than 100microseconds. In some embodiments, the initializing step includesapplying a magnetic field to the plurality of superconducting qubits inthe direction of a vector that is perpendicular to a plane defined bythe plurality of superconducting qubits. In some embodiments, asuperconducting qubit in the plurality of superconducting qubits is apersistent current qubit.

A third aspect of the invention provides a method of determining aquantum state of a first target superconducting qubit. The methodcomprises presenting a plurality of superconducting qubits including afirst target superconducting qubit in the plurality of superconductingqubits. A problem Hamiltonian describes (i) the quantum state of theplurality of superconducting qubits and (ii) each coupling energybetween qubits in the plurality of qubits. The problem Hamiltonian is ator near a ground state. An rf-flux is added to the first targetsuperconducting qubit. The rf-flux has an amplitude that is less thanone flux quantum. An amount of an additional flux in the first targetsuperconducting qubit is adiabatically varied. A presence or an absenceof a dip in a voltage response of a tank circuit that is inductivelycoupled with the first target superconducting qubit during theadiabatically varying step is observed thereby determining the quantumstate of the first target superconducting qubit.

In some embodiments in accordance with this third aspect of theinvention, each superconducting qubit in the plurality ofsuperconducting qubits is in a quantum ground state during all or aportion of the adding step, the adiabatically varying step, and theobserving step. In some embodiments, the problem Hamiltonian correspondsto a terminus of an adiabatic evolution of the plurality ofsuperconducting qubits. In some embodiments, the method furthercomprises biasing all or a portion of the superconducting qubits in theplurality of superconducting qubits. The problem Hamiltonian furtherdescribes a biasing on the first target superconducting qubit. In someembodiments, an energy of the biasing step exceeds the tunneling energyof a tunneling element of the Hamiltonian of the first targetsuperconducting qubit, thereby causing tunneling to be suppressed in thefirst target superconducting qubit during an instance of the biasingstep, adding step and the adiabatically varying step.

In some embodiments in accordance with this third aspect of theinvention, the method further comprises adiabatically removingadditional flux that was added to the first target superconducting qubitduring the adiabatically varying step. In some embodiments, theadiabatically varying step comprises adiabatically varying theadditional flux in accordance with a waveform selected from the groupconsisting of periodic, sinusoidal, triangular, and trapezoidal. In someembodiments, the adiabatically varying step comprises adiabaticallyvarying the additional flux in accordance with a low harmonic Fourierapproximation of a waveform selected from the group consisting ofperiodic, sinusoidal, triangular, and trapezoidal. In some embodiments,the additional flux has a direction that is deemed positive or negative.In some embodiments, the adiabatically varying step is characterized bya waveform that has an amplitude that grows with time. The amplitude ofthe waveform corresponds to an amount of additional flux that is addedto the first target superconducting qubit during the adiabaticallyvarying step. In some embodiments, the additional flux has anequilibrium point that varies with time. In some embodiments, theadditional flux is either unidirectional or bidirectional. In someembodiments, the additional flux has a frequency of oscillation betweenabout 1 cycle per second and about 100 kilocycles per second.

In some embodiments in accordance with the third aspect of theinvention, the adding step comprises adding the rf-flux using (i) anexcitation device that is inductively coupled to the first targetsuperconducting qubit or (ii) a the tank circuit. In some embodiments,the method further comprises repeating the adding step and theadiabatically varying step between 1 time and 100 times. In suchembodiments, the presence or absence of the dip in the voltage responseof the tank circuit is observed as an average of the voltage response ofthe tank circuit across each instance of the adiabatically varying step.

In some embodiments in accordance with the third aspect of theinvention, the first target superconducting qubit is flipped from anoriginal basis state to an alternate basis state during theadiabatically varying step. The method further comprises returning thefirst target superconducting qubit to its original basis state byadiabatically removing additional flux in the qubit after theadiabatically varying step. In some embodiments, the adiabaticallyvarying step does not alter the quantum state of each of superconductingqubits in the plurality of superconducting qubits other than the firsttarget superconducting qubit. In some embodiments, the method furthercomprises recording a presence or an absence of the dip in the voltageresponse of the tank circuit.

In some embodiments in accordance with the third aspect of theinvention, the method further comprises adding a second rf-flux to asecond target superconducting qubit in the plurality of superconductingqubits. The second rf-flux has an amplitude that is less than one fluxquantum. Then an amount of a second additional flux in the second targetsuperconducting qubit is adiabatically varied. A presence or an absenceof a second dip in a voltage response of a tank circuit that isinductively coupled with the second target superconducting qubit duringsaid adiabatically varying is observed, thereby determining the quantumstate of the second target superconducting qubit.

In some embodiments in accordance with the third aspect of theinvention, the method further comprises designating a differentsuperconducting qubit in the plurality of superconducting qubits as thefirst target superconducting qubit. The adding step and theadiabatically varying step are then reperformed with the differentsuperconducting qubit as the first target superconducting qubit. Thedesignating and reperforming are repeated until all or a portion (e.g.,most, almost all, at least eighty percent) of the superconducting qubitsin the plurality of superconducting qubits has been designated as thefirst target superconducting qubit.

In some embodiments in accordance with the third aspect of theinvention, a tank circuit is inductively coupled with the first targetsuperconducting qubit. The method further comprises performing anadiabatic quantum computation step for an amount of time with theplurality of superconducting qubits prior to the adding step. The amountof time is determined by a factor the magnitude of which is a functionof a number of qubits in the plurality of superconducting qubits. Anamount of an additional flux in the first target superconducting qubitis adiabatically varied. Then, a presence or an absence of a dip in thevoltage response of a tank circuit during the adiabatically varying stepis observed, thereby determining the quantum state of the first targetsuperconducting qubit. In some embodiments, the presence of a dip in thevoltage response of the tank circuit corresponds to the first targetsuperconducting qubit being in a first basis state. The absence of a dipin the voltage response of the tank circuit corresponds to the targetsuperconducting qubit being in a second basis state.

In some embodiments in accordance with the third aspect of theinvention, the adiabatically varying step further comprises identifyingan equilibrium point for the additional flux using an approximateevaluation method. In some embodiments, the method further comprisesclassifying the state of the first target qubit as being in the firstbasis state when the dip in the voltage across the tank circuit occursto the left of the equilibrium point and classifying the state of thefirst target qubit as being in the second basis state when the dip inthe voltage across the tank circuit occurs to the right of theequilibrium point.

A fourth aspect of the present invention comprises a method foradiabatic quantum computing using a quantum system comprising aplurality of superconducting qubits. The quantum system is capable ofbeing in any one of at least two quantum configurations at any givetime. The at least two quantum configurations include a firstconfiguration described by an initialization Hamiltonian H_(O) and asecond configuration described by a problem Hamiltonian H_(P.) TheHamiltonian H_(P) has a ground state. The method comprises initializingthe quantum system to the first configuration. Then the quantum systemis adiabatically changed until it is described by the ground state ofthe problem Hamiltonian H_(P). Then the state of the quantum system isread out.

In some embodiments in accordance with the fourth aspect of theinvention, each respective first superconducting qubit in the pluralityof superconducting qubits is arranged with respect to a respectivesecond superconducting qubit in the plurality of superconducting qubitssuch that the first respective superconducting qubit and thecorresponding second respective superconducting qubit define apredetermined coupling strength. The predetermined coupling strengthbetween each of the first respective superconducting qubits andcorresponding second respective superconducting qubits in the pluralityof superconducting qubits collectively define a computational problem tobe solved. In some instances, the problem Hamiltonian H_(P) comprises atunneling term for each of the respective superconducting qubits in theplurality of superconducting qubits. The energy of the tunneling termfor each respective superconducting qubit in the plurality ofsuperconducting qubits is less than the average of the predeterminedcoupling strengths between each of the first respective superconductingqubits and second respective superconducting qubits in the plurality ofsuperconducting qubits.

In some embodiments in accordance with the fourth aspect of theinvention, the reading out step comprises probing an observable of theσ_(X) Pauli matrix operator or σ_(Z) Pauli matrix operator. In someembodiments, a tank circuit is in inductive communication with all or aportion of the superconducting qubits in the plurality ofsuperconducting qubits. In such embodiments, the reading out stepcomprises measuring a voltage across the tank circuit. In someembodiments, the superconducting qubit in the plurality ofsuperconducting qubits is a persistent current qubit.

A fifth aspect of the present invention provides a structure foradiabatic quantum computing comprising a plurality of superconductingqubits. The plurality of superconducting qubits are capable of being inany one of at least two configurations at any give time. The at leasttwo configurations include a first configuration characterized by aninitialization Hamiltonian H₀ and a second Hamiltonian characterized bya problem Hamiltonian H_(P). The problem Hamiltonian has a ground state.Each respective first superconducting qubit in the plurality ofsuperconducting qubits is coupled with a respective secondsuperconducting qubit in the plurality of superconducting qubits suchthat the first respective superconducting qubit and the correspondingsecond respective superconducting qubit define a predetermined couplingstrength. The predetermined coupling strength between each of the firstrespective superconducting qubits and the corresponding secondrespective superconducting qubits collectively define a computationalproblem to be solved. A tank circuit is inductively coupled to all or aportion of the plurality of superconducting qubits.

In some embodiments in accordance with the fifth aspect of theinvention, a superconducting qubit in the plurality of superconductingqubits is a persistent current qubit. In some embodiments, the tankcircuit has a quality factor that is greater than 1000. In someembodiments, the tank circuit comprises an inductive element. Theinductive element comprises a pancake coil of superconducting material.In some embodiments, the pancake coil of a superconducting materialcomprising a first turn and a second turn. The superconducting materialof the pancake coil is niobium. Furthermore, there is a spacing of 1about micrometer between the first turn and the second turn of thepancake coil.

In some embodiments in accordance with the fifth aspect of theinvention, the tank circuit comprises an inductive element and acapacitive element that are arranged in parallel or in series withrespect to each other. In some embodiments, the tank circuit comprisesan inductive element and a capacitive element that are arranged inparallel with respect to each other and the tank circuit has aninductance between about 50 nanohenries and about 250 nanohenries. Insome embodiments, the tank circuit comprises an inductive element and acapacitive element that are arranged in parallel with respect to eachother and the tank circuit has a capacitance between about 50 picofaradsand about 2000 picofarads. In some embodiments, the tank circuitcomprises an inductive element and a capacitive element that arearranged in parallel with respect to each other and the tank circuit hasa resonance frequency between about 10 megahertz and about 20 megahertz.In some embodiments, the tank circuit has a resonance frequency f_(T)that is determined by the equality:f _(T)=ω_(T)/2π=1/{square root}{square root over (L _(T) C _(T))}such that

-   -   L_(T) is an inductance of the tank circuit; and    -   C_(T) is a capacitance of the tank circuit.

In some embodiments in accordance with the fifth aspect of theinvention, the tank circuit comprises one or more Josephson junctions.In some embodiments, the structure further comprises means for biasingthe one or more Josephson junctions of the tank circuit. In someembodiments, the structure further comprises an amplifier connectedacross the tank circuit in such a manner that the amplifier can detect achange in voltage across the tank circuit. In some embodiments, theamplifier comprises a high electron mobility field-effect transistor(HEMT) or a pseudomorphic high electron mobility field-effect transistor(PHEMT). In some embodiments, the amplifier comprises a multi-stageamplifier. In some embodiments, the multi-stage amplifier comprises two,three, or four transistors In some embodiments, structure furthercomprises a helium-3 pot of a dilution refrigerator that is thermallycoupled to all or a portion of the plurality of superconducting qubits.to.

In some embodiments in accordance with the fifth aspect of theinvention, the structure further comprising means for driving the tankcircuit by a direct bias current I_(DC). In some embodiments, thestructure further comprises means for driving the tank circuit by analternating current I_(RF) of a frequency ω close to the resonancefrequency ω₀ of the tank circuit. In some embodiments, the totalexternally applied magnetic flux to a superconducting qubit in theplurality of superconducting qubits, Φ_(E), isΦ_(E)=Φ_(DC)+Φ_(RF)where,

-   -   Φ_(RF) is an amount of applied magnetic flux contributed to the        superconducting qubit by the alternating current I_(RF); and    -   Φ_(DC) is an amount of applied magnetic flux that is determined        by the direct bias current I_(DC). In some embodiments, the        structure further comprises means for applying a magnetic field        on the superconducting qubit, and wherein        Φ_(DC)=Φ_(A) +f(t)Φ₀,        where,    -   Φ₀ is one flux quantum;    -   f(t)Φ₀ is constant or is slowly varying and is generated by the        direct bias current I_(DC); and        Φ_(A) =B _(A) ×L _(Q),        such that    -   B_(A) is a magnitude of the magnetic field applied on the        superconducting qubit by the means for applying the magnetic        field; and    -   L_(Q) is an inductance of the superconducting qubit.        In some embodiments f(t) has a value between 0 and. In some        embodiments, the means for applying a magnetic field on the        superconducting qubit comprises a bias line that is magnetically        coupled to the superconducting qubit. In some embodiments, the        means for applying a magnetic field on the superconducting qubit        is an excitation device. In some embodiments, Φ_(RF) has a        magnitude between about 10⁻⁵Φ₀ and about 10⁻¹Φ₀. In some        embodiments, the structure further comprises means for varying        f(t), Φ_(A), and/or Φ_(RF). In some embodiments, the structure        further comprises means for varying Φ_(RF) in accordance with a        small amplitude fast function. In some embodiments, the means        for varying Φ_(RF) in accordance with a small amplitude fast        function is a microwave generator that is in electrical        communication with the tank circuit.

In some embodiments in accordance with the fifth aspect of theinvention, the structure further comprises an amplifier connected acrossthe tank circuit and means for measuring a total impedance of the tankcircuit, expressed through the phase angle χ between driving currentI_(RF) and the tank voltage. In some embodiments, the means formeasuring a total impedance of the tank circuit is an oscilloscope.

A sixth aspect of the invention provides a computer program product foruse in conjunction with a computer system. The computer program productcomprises a computer readable storage medium and a computer programmechanism embedded therein. The computer program mechanism comprisesinstructions for initializing a quantum system comprising a plurality ofsuperconducting qubits to an initialization Hamiltonian H_(O). Thequantum system is capable of being in one of at least two configurationsat any give time. The at least two configurations include a firstconfiguration characterized by the initialization Hamiltonian H_(O) anda second configuration characterized by a problem Hamiltonian H_(P).Each respective first superconducting qubit in the plurality ofsuperconducting qubits is arranged with respect to a respective secondsuperconducting qubit in the plurality of superconducting qubits suchthat the first respective superconducting qubit and the secondrespective superconducting qubit define a predetermined couplingstrength. The predetermined coupling strengths between each of the firstrespective superconducting qubits and the second respectivesuperconducting qubits collectively define a computational problem to besolved. The computer program mechanism further comprises instructionsfor adiabatically changing the quantum system until it is described bythe ground state of the problem Hamiltonian H_(P) and instructions forreading out the state of the quantum system.

In some embodiments in accordance with this sixth aspect of theinvention, the computer program mechanism further comprises instructionsfor repeating the instructions for biasing, instructions for adding, andinstructions for adiabatically varying between 1 time and 100 timesinclusive. The presence or absence of the voltage response of the tankcircuit is observed as an average of the voltage response of the tankcircuit to each instance of the instructions for adiabatically changingthat are executed by the instructions for repeating.

A seventh aspect of the invention comprises a computer program productfor use in conjunction with a computer system. The computer programproduct comprises a computer readable storage medium and a computerprogram mechanism embedded therein. The computer program mechanismdetermines a quantum state of a first target superconducting qubit in aplurality of superconducting qubits. The computer program mechanismcomprises instructions for initializing a plurality of superconductingqubits so that they are described by a problem Hamiltonian. The problemHamiltonian describes (i) the quantum state of the plurality ofsuperconducting qubits and (ii) each coupling energy between qubits inthe plurality of qubits. The problem Hamiltonian is at or near a groundstate. The computer program mechanism further comprises instructions foradding an rf-flux to the first target superconducting qubit. The rf-fluxhas an amplitude that is less than one flux quantum. The computerprogram mechanism further comprises instructions for adiabaticallyvarying an amount of an additional flux in the first targetsuperconducting qubit and observing a presence or an absence of a dip ina voltage response of a tank circuit that is inductively coupled withthe first target superconducting qubit during the adiabatically varyingstep.

In some embodiments in accordance with this seventh aspect of theinvention, each superconducting qubit in the plurality ofsuperconducting qubits is in a quantum ground state during all or aportion of the instructions for initializing, instructions for adding,and the instructions for adiabatically varying. In some embodiments, theproblem Hamiltonian corresponds to a terminus of an adiabatic evolutionof the plurality of superconducting qubits. In some embodiments, thecomputer program product further comprises instructions for biasing allor a portion of the superconducting qubits in the plurality ofsuperconducting qubits. In such embodiments, the problem Hamiltonianadditionally describes the biasing on the qubits in the plurality ofsuperconducting qubits. In some embodiments, an energy of the biasingexceeds the tunneling energy of a tunneling element of the Hamiltonianof a superconducting qubit in the plurality of superconducting qubitsthereby causing tunneling to be suppressed in the superconducting qubitduring an instance of the instructions for biasing, instructions foradding and the instructions for adiabatically varying.

In some embodiments in accordance with the seventh aspect of theinvention, the computer program mechanism further comprises instructionsfor adiabatically removing additional flux that was added to the firsttarget superconducting qubit during the instructions for adiabaticallyvarying. In some embodiments, the instructions for adiabatically varyingcomprise instructions for adiabatically varying the additional flux inaccordance with a waveform selected from the group consisting ofperiodic, sinusoidal, triangular, and trapezoidal. In some embodiments,the instructions for adiabatically varying comprise instructions foradiabatically varying the additional flux in accordance with a lowharmonic Fourier approximation of a waveform selected from the groupconsisting of periodic, sinusoidal, triangular, and trapezoidal. In someembodiments, the additional flux has a direction that is deemed positiveor negative. In some embodiments, the instructions for adiabaticallyvarying are characterized by a waveform that has an amplitude that growswith time and such that the amplitude of the waveform corresponds to anamount of additional flux that is added to the first targetsuperconducting qubit during an instance of the instructions foradiabatically varying.

In some embodiments in accordance with the seventh aspect of theinvention, the additional flux has an equilibrium point that varies withtime. In some embodiments, the additional flux is either unidirectionalor bidirectional. In some embodiments, the additional flux has afrequency of oscillation between about 1 cycle per second and about 100kilocycles per second. In some embodiments, the instructions for addingcomprise instructions for adding the rf-flux using (i) an excitationdevice that is inductively coupled to the first target superconductingqubit or (ii) the tank circuit. In some embodiments, the computerprogram mechanism further comprises instructions for repeating theinstructions for adding and the instructions for adiabatically varyingbetween 1 time and 100 times. In such embodiments, the presence orabsence of the voltage response of the tank circuit is observed as anaverage of the voltage response of the tank circuit across each instanceof the instructions for adiabatically varying that is executed by theinstructions for repeating.

An eight aspect of the invention comprises a computer system fordetermining a quantum state of a first target superconducting qubit in aplurality of superconducting qubits. The computer system comprises acentral processing unit and a memory, coupled to the central processingunit. The memory stores instructions for biasing all or a portion of thequbits in the plurality of superconducting qubits other than the firsttarget superconducting qubit. A problem Hamiltonian describes (i) thebiasing on the qubits in the plurality of superconducting qubits and(ii) each coupling energy between respective superconducting qubit pairsin the plurality of superconducting qubits. The problem Hamiltonian isat or near a ground state. The memory further stores instructions foradding an rf-flux to the first target superconducting qubit. The rf-fluxhas an amplitude that is less than one flux quantum. The memory furtherstores instructions for adiabatically varying an amount of an additionalflux in the first target superconducting qubit and observing a presenceor an absence of a dip in a voltage response of a tank circuit that isinductively coupled with the first target superconducting qubit during atime when the instructions for adiabatically varying are executed.

A ninth aspect of the present invention provides a computation devicefor adiabatic quantum computing comprising a plurality ofsuperconducting qubits. Each superconducting qubit in the plurality ofsuperconducting qubits comprises two basis states associated with theeigenstates of a σ^(Z) Pauli matrix that can be biased. The quantumcomputation device further comprises a plurality of couplings. Eachcoupling in the plurality of couplings is disposed between asuperconducting qubit pair in the plurality of superconducting qubits.Each term Hamiltonian for a coupling in the plurality of couplings has aprincipal component proportional to σ^(Z){circle over (×)}σ^(Z). Thesign for at least one principal component proportional to σ^(Z){circleover (×)}σ^(Z) for a coupling in the plurality of couplings isantiferromagnetic. The superconducting qubits and the plurality ofcouplings are collectively capable of being in any one of at least twoconfigurations. The at least two configurations include a firstconfiguration characterized by an initialization Hamiltonian H₀ and asecond Hamiltonian characterized by a problem Hamiltonian H_(P). Theproblem Hamiltonian has a ground state. Each respective firstsuperconducting qubit in the plurality of superconducting qubits iscoupled with a respective second superconducting qubit in the pluralityof superconducting qubits such that the first respective superconductingqubit and the corresponding second respective superconducting qubitdefine a predetermined coupling strength. The predetermined couplingstrength between each of the first respective superconducting qubits andthe corresponding second respective superconducting qubits collectivelydefine a computational problem to be solved. The computation devicefurther comprises a read out circuit coupled to at least onesuperconducting qubit in the plurality of superconducting qubits.

A tenth aspect of the invention comprises an apparatus comprising aplurality of superconducting charge qubits. Each respective firstsuperconducting charge qubit in the plurality of superconducting chargequbits is coupled with a respective second superconducting charge qubitin the plurality of superconducting charge qubits such that the firstrespective superconducting charge qubit and the second respectivesuperconducting charge qubit define a predetermined coupling strength.The predetermined coupling strength between each of the first respectivesuperconducting charge qubits and each of the second respectivesuperconducting charge qubits in the plurality of superconducting chargequbits collectively define a computational problem to be solved. Eachsuperconducting charge qubit in the plurality of superconducting chargequbits is capable of being in one of at least two configurations. Theseat least two configurations include a first configuration in accordancewith an initialization Hamiltonian H₀ and a second configuration inaccordance with a problem Hamiltonian H_(P). The apparatus furthercomprises an electrometer coupled to a superconducting charge qubit inthe plurality of superconducting charge qubits.

In some embodiments in accordance with this tenth aspect of theinvention, a superconducting charge qubit in the plurality ofsuperconducting charge qubits comprises (i) a mesoscopic island made ofsuperconducting material, (ii) superconducting reservoir, and (iii) aJosephson junction connecting the mesoscopic island to thesuperconducting reservoir. In some embodiments, the Josephson junctionis a split Josephson junction. In some embodiments, the superconductingcharge qubit further comprises a flux source configured to apply flux tothe split Josephson junction.

In some embodiments in accordance with the tenth aspect of theinvention, the apparatus further comprises a generator capacitivelycoupled to a superconducting charge qubit in the plurality ofsuperconducting charge qubits by a capacitor. In some embodiments, thegenerator is configured to apply a plurality of electrostatic pulses tothe superconducting charge qubit. The plurality of electrostatic pulsesadditionally define the computational problem.

In some embodiments in accordance with the tenth aspect of theinvention, the apparatus further comprises a variable electrostatictransformer disposed between a first superconducting charge qubit and asecond superconducting charge qubit in the plurality of superconductingcharge qubits such that the predetermined coupling strength between thefirst superconducting charge qubit and the second superconducting chargequbit is tunable. In some embodiments, each respective firstsuperconducting charge qubit in the plurality of superconducting chargequbits is arranged with respect to a respective second superconductingcharge qubit in the plurality of superconducting charge qubits such thatthe plurality of superconducting charge qubits collectively form anon-planar graph.

An eleventh aspect of the invention provides a method for computingusing a quantum system comprising a plurality of superconducting chargequbits. The quantum system is coupled to an electrometer and the quantumsystem is capable of being in any one of at least two configurations.The at least two configurations includes a first configurationcharacterized by an initialization Hamiltonian H₀ and a secondconfiguration characterized by a problem Hamiltonian H_(P). The problemHamiltonian has a ground state. The plurality of superconducting chargequbits are arranged with respect to one another, with a predeterminednumber of couplings between respective pairs of superconducting chargequbits in the plurality of charge qubits, such that the plurality ofsuperconducting charge qubits, coupled by the predetermined number ofcouplings, collectively define a computational problem to be solved. Themethod comprises initializing the quantum system to the initializationHamiltonian H_(O). Then the quantum system is adiabatically changeduntil it is described by the ground state of the problem HamiltonianH_(P). Next the quantum state of each superconducting charge qubit inthe quantum system is read out through the electrometer, thereby solvingthe computational problem to be solved.

In some embodiments in accordance with the eleventh aspect of theinvention, a first superconducting charge qubit in the plurality ofsuperconducting charge qubits is coupled to a second superconductingcharge qubit in the plurality of superconducting charge qubits by acapacitor such that the predetermined coupling strength between thefirst superconducting charge qubit and the second superconducting chargequbit is predetermined and is a function of the physical properties ofthe capacitor.

In some embodiments in accordance with the eleventh aspect of theinvention, a first superconducting charge qubit in the plurality ofsuperconducting charge qubits is coupled to a generator by a deviceconfigured to provide a tunable effective charging energy. The devicecomprises a capacitor and the method further comprises: tuning the valueof the effective charging energy of the first superconducting chargequbit by varying the gate voltage on the capacitor of said device. Insome embodiments, a superconducting charge qubit in the plurality ofsuperconducting charge qubits comprises a split Josephson junctionhaving a variable effective Josephson energy. In such embodiments, themethod further comprises tuning the value of the effective Josephsonenergy of the superconducting charge qubit by varying a flux applied tothe split Josephson junction. In some embodiments, the firstconfiguration is reached by setting the effective Josephson energy ofthe superconducting charge qubit to a maximum value.

In some embodiments in accordance with the eleventh aspect of theinvention, the adiabatically changing step comprises changing theconfiguration of the system from the first configuration characterizedby the initialization Hamiltonian H₀, to the second Hamiltoniancharacterized by a problem Hamiltonian H_(P) in the presence oftunneling on a superconducting charge qubit in the plurality ofsuperconducting charge qubits.

In some embodiments in accordance with an eleventh aspect of theinvention, a first superconducting charge qubit in the plurality ofsuperconducting charge qubits is characterized by (i) an effectiveJosephson energy that is tunable and (ii) an effective charging energythat is tunable. A minimum value of the effective Josephson energy isless than the effective charging energy of the first superconductingcharge qubit. A minimum value of the effective Josephson energy is lessthan a strength of a coupling between the first superconducting chargequbit and a second superconducting charge qubit in the plurality ofsuperconducting charge qubits. The effective charging energy is, atmost, equal to a maximum value of the effective Josephson energy of thefirst superconducting charge qubit. Furthermore, a strength of acoupling between the first superconducting charge qubit and a secondsuperconducting charge qubit in the plurality of superconducting chargequbits is, at most, equal to a maximum value of the effective Josephsonenergy of the first superconducting charge qubit.

In still another embodiment in accordance with the eleventh aspect ofthe invention, a first superconducting charge qubit in the plurality ofsuperconducting charge qubits is characterized by (i) an effectiveJosephson energy that is tunable and (ii) an effective charging energythat is tunable. In such embodiments, the adiabatically changing stepcomprises adiabatically tuning the effective Josephson energy of thefirst superconducting charge qubit such that the effective Josephsonenergy of the first superconducting charge qubit reaches a minimum valuewhen the quantum system is described by the ground state of the problemHamiltonian H_(P).

In some embodiments in accordance with the eleventh aspect of theinvention, a first superconducting charge qubit in the plurality ofsuperconducting charge qubits has a first basis state and a second basisstate and, when the quantum system is described by the ground state ofthe problem Hamiltonian H_(P), tunneling between the first basis stateand the second basis state of the first superconducting charge qubitdoes not occur.

In some embodiments in accordance with the eleventh aspect of theinvention, a first superconducting charge qubit in the plurality ofsuperconducting charge qubits has a first basis state and a second basisstate and, when the quantum system is described by the ground state ofthe problem Hamiltonian H_(P), the tunneling between the first basisstate and the second basis state of the first superconducting chargequbit does occur. Furthermore, the reading out step comprises probing anobservable of the sigma-x Pauli matrix σ^(X).

In some embodiments in accordance with the eleventh aspect of theinvention, a first superconducting charge qubit in the plurality ofsuperconducting charge qubits is characterized by (i) an effectiveJosephson energy that is tunable and (ii) an effective charging energythat is tunable. In such embodiments, a minimum value of the effectiveJosephson energy is less than the effective charging energy of the firstsuperconducting charge qubit; a minimum value of effective Josephsonenergy is less than a strength of a coupling between the firstsuperconducting charge qubit and a second superconducting charge qubitin the plurality of superconducting charge qubits; the effectivecharging energy is greater than a maximum value of the effectiveJosephson energy of the first superconducting charge qubit; and astrength of a coupling between the first superconducting charge qubitand a second superconducting charge qubit in the plurality ofsuperconducting charge qubits is, at most, equal to the maximumeffective Josephson energy of the first superconducting charge qubit. Insome such embodiments, the initializing step comprises setting theeffective charging energy of the first superconducting charge qubit to aminimum value. In some such embodiments, the adiabatically changing stepcomprises adiabatically tuning the effective Josephson energy of thefirst superconducting charge qubit such that the effective Josephsonenergy is at a minimum value when the quantum system is described by theground state of the problem Hamiltonian H_(P), and adiabaticallyincreasing the effective charging energy of the first superconductingcharge qubit.

In some embodiments in accordance with the eleventh aspect of theinvention, a first superconducting charge qubit in the plurality ofsuperconducting charge qubits is characterized by an effective Josephsonenergy that is tunable. The initializing step comprises setting theeffective Josephson energy of the first superconducting charge qubit toa minimum value, and the adiabatically changing step comprises (i)adiabatically tuning the effective Josephson energy of the firstsuperconducting charge qubit such that the effective Josephson energy isgreater than a minimum value for a period of time before the quantumsystem is described by the ground state of the problem HamiltonianH_(P), and adiabatically tuning the effective Josephson energy of thefirst superconducting charge qubit such that the effective Josephsonenergy is at a minimum value when the quantum system is described by theground state of the problem Hamiltonian H_(P).

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A illustrates a known superconducting qubit.

FIG. 1B illustrates a known energy potential for a superconductingqubit.

FIG. 2 illustrates an exemplary quantum computation circuit model inaccordance with the prior art.

FIG. 3 illustrates a known general equation that describes the theory ofadiabatic quantum computing.

FIG. 4 illustrates a work flow diagram for a process of adiabaticquantum computing.

FIGS. 5A-5G illustrates arrangements of superconducting qubits foradiabatic quantum computing in accordance with some embodiments of thepresent invention.

FIGS. 6A-6B illustrates an example of a computational problem that canbe solved by adiabatic quantum computing.

FIG. 7A illustrates an energy level diagram for a system comprising aplurality of superconducting qubits during an instantaneous adiabaticchange of the system.

FIG. 7B illustrates an energy level diagram for a system comprising aplurality of superconducting qubits when the plurality of qubits aredescribed by the ground state of a problem Hamiltonian H_(P).

FIG. 8 illustrates a device for controlling and reading out the state ofa superconducting qubit for adiabatic quantum computing in accordancewith some embodiments of the present invention.

FIG. 9A illustrates an energy level diagram for a physical system inwhich the energy level diagram exhibits an anticrossing between energylevels of the physical system in accordance with an embodiment of thepresent invention.

FIG. 9B illustrates an energy level diagram for a physical system havingan energy level crossing in accordance with an embodiment of the presentinvention.

FIGS. 10A-10D illustrate the waveforms for additional flux Φ_(A) in aqubit undergoing adiabatic quantum change, in arbitrary units, inaccordance with various embodiments of the present invention.

FIG. 11A illustrates the form of a readout signal for a superconductingqubit having an anticrossing between two energy levels.

FIG. 11B illustrates the form of a readout signal for a superconductingqubit having that does not have anticrossing between two energy levels.

FIGS. 12A-12D illustrate superconducting charge qubits and read outdevices in accordance with some embodiments of the present invention.

FIGS. 13A-13B illustrate coupled superconducting charge qubits inaccordance with some embodiments of the present invention.

FIGS. 14A-14D illustrates how superconducting qubits can be arranged inaccordance some embodiments of the present invention.

FIG. 15 illustrates a system that is operated in accordance with someembodiments of the present invention.

Like reference numerals refer to corresponding parts throughout theseveral views of the drawings.

DETAILED DESCRIPTION OF THE INVENTION

The present invention comprises systems and methods for adiabaticquantum computing using superconducting qubits. In various embodimentsof the present invention, adiabatic quantum computing is performed onregisters of superconducting qubits that have demonstrated quantumcomputing functionality. Adiabatic quantum computing is a model ofquantum computing that can be used to attempt to find solutions forcomputationally difficult problems.

General Embodiments

When choosing a candidate system for adiabatic quantum computing thereare a few criteria that can be observed. These criteria can be drawnfrom those described herein below. However, some embodiments of thepresent invention may not adhere to all of these criteria. One criterionis that the readout device should a Stem-Gerlach σ^(Z) type observation.A second criterion is that the tunneling term in the problem Hamiltonianshould be about zero. For H_(P)=Δσ^(X)+εσ^(Z) then Δ≈0. A thirdcriterion is that the magnitude of the tunneling term in the problem,initial, or extra Hamiltonian (H_(P), H₀, H_(E)) should be tunable. Afourth criterion is that the qubit-qubit coupling should be diagonal inthe basis of final qubit states, i.e., σ^(Z){circle over (×)}σ^(Z).Because an Ising model with ferromagnetic couplings has a trivial groundstate, all spins aligned, a fifth criterion is that the system have someantiferromagnetic coupling between qubits. Some AFM couplings includethe case where all are antiferromagnetic. Also, ferromagnetic couplingshave a negative sign −Jσ^(Z){circle over (×)}σ^(Z), andantiferromagnetic couplings have a positive sign Jσ^(Z){circle over(×)}σ^(Z).

Some embodiments of the present invention adhere to the above criteria.Other embodiments of the present invention do not. For instance, in thecase of the phase qubit, it is possible to have the tunneling term inthe problem Hamiltonian be, not zero, but weak, e.g., forH_(P)=Δσ^(X)+εσ^(Z) then Δ<<ε. In such a case it is possible for thereadout device to probe a Stem-Gerlach σ^(X) type observable. Otherembodiments of superconducting adiabatic quantum computers of thepresent invention do not adhere to the third criterion described above.For example, the magnitude of the tunneling term in the problem,initial, or extra Hamiltonian (H_(P), H₀, H_(E)) is fixed but thecontribution of the problem, initial, or extra Hamiltonian to theinstant Hamiltonian is tunable in such embodiments. Specific embodimentsof the present invention are described below.

5.1 Exemplary General Procedure

In accordance with embodiments of the present invention, the generalprocedure of adiabatic quantum computing is shown in FIG. 4. In step401, a quantum system that will be used to solve a computation isselected and/or constructed. In some embodiments, each problem or classof problems to be solved requires a custom quantum system designedspecifically to solve the problem. Once a quantum system has beenchosen, an initial state and a final state of the quantum system need tobe defined. The initial state is characterized by the initialHamiltonian H₀ and the final state is characterized by the finalHamiltonian H_(P) that encodes the computational problem to be solved.In preferred embodiments, the quantum system is initiated to the groundstate of the initial Hamiltonian H₀and, when the system reaches thefinal state, it is in the ground state of the final Hamiltonian H_(P).More details on how systems are selected and designed to solve acomputational problem are described below.

In step 403, the quantum system is initialized to the ground state ofthe time-independent Hamiltonian, H₀, which initially describes thequantum system. It is assumed that the ground state of H₀ is a state towhich the system can be reliably and reproducibly set. As will bedisclosed in further detail below, this assumption is reasonable forquantum systems comprising, at a minimum, specific types of qubits andspecific types of arrangements of such qubits.

In transition 404 between steps 403 and 405, the quantum system is actedupon in an adiabatic manner in order to alter the system. The systemchanges from being described by Hamiltonian H₀ to a description underH_(P). This change is adiabatic, as defined above, and occurs in aperiod of time T. In other words, the operator of an adiabatic quantumcomputer causes the system, and Hamiltonian H describing the system, tochange from H₀ to a final form H_(P) in time T. The change is aninterpolation between H₀ and H_(P). The change can be a linearinterpolation:H(t)=(1−γ(t))H ₀+γ(t)H _(P)where the adiabatic evolution parameter, γ(t), is a continuous functionwith γ(t=0)=0, and γ(t=T)=1. The change can be a linear interpolation,γ(t)=t/T such that${H\left( \frac{t}{T} \right)} = {{\left( {1 - \frac{t}{T}} \right)\quad H_{0}} + {\frac{t}{T}{H_{P}.}}}$In accordance with the adiabatic theorem of quantum mechanics, a systemwill remain in the ground state of H at every instance the system ischanged and after the change is complete, provided the change isadiabatic. In some embodiments of the present invention, the quantumsystem starts in an initial state H₀ that does not permit quantumtunneling, is perturbed in an adiabatic manner to an intermediate statethat permits quantum tunneling, and then is perturbed in an adiabaticmanner to the final state described above.

In step 405, the quantum system has been altered to one that isdescribed by the final Hamiltonian. The final Hamiltonian H_(P) canencode the constraints of a computational problem such that the groundstate of H_(P) corresponds to a solution to this problem. Hence, thefinal Hamiltonian is also called the problem Hamiltonian H_(P). If thesystem is not in the ground state of H_(P), the state is an approximatesolution to the computational problem. Approximate solutions to manycomputational problems are useful and such embodiments are fully withinthe scope of the present invention.

In step 407, the system described by the final Hamiltonian H_(P) is readout. The read out can be in the σ^(Z) basis of the qubits. If the readout basis commutes with the terms of the problem Hamiltonian H_(P), thenperforming a read out operation does not disturb the ground state of thesystem. The read out method can take many forms. The object of the readout step is to determine exactly or approximately the ground state ofthe system. The states of all qubits are represented by the vector{right arrow over (O)}, which gives a concise image of the ground stateor approximate ground state of the system. The read out method cancompare the energies of various states of the system. More examples aregiven below, making use of specific qubits for better description.

5.2 Changing a Quantum System Adiabatically

In one embodiment of the present invention, the natural quantummechanical evolution of the quantum system under the slowly changingHamiltonian H(t) carries the initial state H₀ of the quantum system intoa final state, the ground state of H_(P), corresponding to the solutionof the problem defined by the quantum system. A measurement of the finalstate of the quantum system reveals the solution to the computationalproblem encoded in the problem Hamiltonian. In such embodiments, anaspect that can define the success of the process is how quickly (orslowly) the change between the initial Hamiltonian and problemHamiltonian occurs. How quickly one can drive the interpolation betweenH₀ and H_(P), while keeping the system in the ground state of theinstantaneous Hamiltonians that the quantum system traverses through inroute to H_(P), is a determination that can be made using the adiabatictheorem of quantum mechanics. This section provides a detailedexplanation of the adiabatic theorem of quantum mechanics. Moreparticularly, this section describes how quantum mechanics imposesconstraints on which quantum systems can be used in accordance with thepresent invention and how such quantum systems can be used to solvecomputational problems using the methods of the present invention.

A quantum system evolves under the Schrödinger equation:${{{\mathbb{i}}\quad\frac{\partial}{\partial t}\left. {\psi(t)} \right\rangle} = {{\Theta(t)}\quad{H(t)}\quad\left. {\psi(t)} \right\rangle}},$where Θ(t) and H(t) are respectively the time ordering operator andHamiltonian of the system. Adiabatic evolution is a special case whereH(t) is a slowly varying function. The time dependent basis states andenergy levels of the Hamiltonian are:H(t)|l;t>=E _(l)(t)|l;t>where l ε [0, N-1], and N is the dimension of the Hilbert space for thequantum system described by the Schrödinger equation. The energy levels,energies, or energy spectra of the quantum system E_(l)(t) are a set ofenergies that the system can occupy. The energies of the states are astrictly increasing set.

A general example of the adiabatic evolution of a quantum system is asfollows. The states |0;t> and |1;t) are respectively the ground andfirst excited states of Hamiltonian H(t), with energies E₀(t) and E₁(t).Gap g(t) is the difference between energies of the ground and firstexcited states as follows:g(t)=E ₁(t)−E ₀(t).If the quantum system is initialized in the ground state and evolvedunder H(t), where H(t) is slowly varying, and if the gap is greater thanzero, then for 0≦t≦T the quantum system will remain in the ground state.In other words:|<E ₀ ;T|ψ(T)>|²≧1−ε².Without intending to be limited to any particular theory, it is believedthat the existence of the gap means that the quantum system under theSchrödinger equation remains in the ground state with high fidelity,e.g., 1−ε² (ε<<1). The fidelity of the operation can be foundquantitatively.

The minimum energy gap between the ground state E₀ and first excitedstate E₁ of the instantaneous Hamiltonian is given by g_(min) where:$g_{\min} = {\min\limits_{0 \leq t \leq T}{\left\lbrack {{E_{1}(t)} - {E_{0}(t)}} \right\rbrack.}}$Also relevant is the matrix element:$\left\langle \frac{\mathbb{d}H}{\mathbb{d}t} \right\rangle_{1,0} = {\left\langle {E_{1};{t\quad{\frac{\mathbb{d}H}{\mathbb{d}t}}\quad E_{0}};t} \right\rangle.}$The adiabatic theorem asserts fidelity of the quantum system will beclose to unity provided that:$\frac{\left\langle \frac{\mathbb{d}H}{\mathbb{d}t} \right\rangle_{1,0}}{g_{\min}^{2}} \leq ɛ$If this criterion is met, the quantum system will remain in the groundstate.

In an embodiment of the present invention, T is the time taken to vary acontrol parameter of a charge qubit, for example induced gate charge orflux for a charge qubit with split Josephson junction. See Section 5.4.In an embodiment of the present invention, time T is a value betweenabout 0.1 nanosecond and about 500 microseconds. In other words, theamount of time between when the quantum system is allowed to beginadiabatically changing from the initial state H₀ to when the quantumsystem first reaches the final state H_(P) is between about 0.1nanosecond and about 500 microseconds. In another embodiment of thepresent invention, time T is a value between about 10 nanosecond andabout 100 microsecond. In an embodiment of the present invention, time Tis a value less than the inverse of the characteristic frequency of thephysical system comprising superconducting qubits. The characteristicfrequency of a qubit is the energy difference between a ground state andan excited state of a qubit, converted to energy units. Thus, thecharacteristic frequency of a physical system comprising qubits is thecharacteristic frequency of one or more qubits within the physicalsystem.

In an embodiment of the present invention, T is the time taken to vary acontrol parameter of a phase qubit, for example flux in a persistentcurrent qubit. See Section 5.3. In an embodiment of the presentinvention, time T is a value between about 1 nanosecond and about 100microseconds. In other words, the amount of time between when thequantum system is allowed to begin adiabatically changing from theinitial state H₀ to when the quantum system first reaches the finalstate H_(P) is between about 1 nanosecond and about 100 microseconds. Inan embodiment of the present invention, time T is a value between about4 nanoseconds and about 10 microseconds. In some embodiments, the time Tis calculated as the time at which a Landau-Zener transition is likelynot to occur. For more information on Landau-Zener transitions see, forexample, Garanin et al., 2002, Phys. Rev. B 66, 174438, which is herebyincorporated by reference in its entirety.

5.2.1 Application of Landau-Zener Theory

Most analyses of adiabatic algorithms emphasize how the gap, g(t), andthe ratio of the matrix element <dH/dt>_(1,0) to the square of theminimum of the gap, scale with increasing problem size. It is believedthat, by examining these metrics, the validity of adiabatic algorithms,and other adiabatic processes, can be determined. Some embodiments ofthe present invention make use of an alternative analysis that looks atthe probability of transition 404, or some other process, being diabatic(i.e. a process that involves heat transfer as opposed to an adiabaticprocess that involves no heat transfer). Rather than calculating theminimum gap, which is the difference between the energy of the groundand the first excited state of the quantum system 850 that models theproblem to be solved, this additional analysis calculates theprobability of a transition out of the ground state by the quantumsystem. In examples of the present invention, this probabilitycalculation can be a more relevant metric for assessing the failure rateof the adiabatic algorithm or process. To perform the computation, it isassumed that any diabatic transition (any transition characterized bythe transfer of heat) is a Landau-Zener transition, e.g., a transitionconfined to adjacent levels at anticrossings. A description ofanticrossing levels is provided below in conjunction with FIGS. 9A and9B. When the state of a plurality of qubits approaches an anticrossing,the probability for a transition out of the ground state can beparameterized by (i) the minimum of the gap, g_(min), (ii) thedifference in the slopes, Δ_(m), of the asymptotes for the energy levelsof the qubit or plurality of qubits undergoing adiabatic change (e.g.,quantum system 850 of FIG. 8), and (iii) the rate of change of theadiabatic evolution parameter, dγ/dt={dot over (γ)}. For moreinformation on the asymptotes for the energy levels of quantum system850, see Section 5.3.2, below. The first estimate of the Landau-Zenertransition probability is: $\begin{matrix}{{P_{LZ} = {\mathbb{e}}^{{- 2}\quad\pi\quad\eta}};} & {\eta = {\frac{1}{4\hslash}{\frac{g_{\min}}{{{\Delta\quad m}}\quad\overset{.}{\gamma}}.}}}\end{matrix}$The values for the parameters g_(min), and Δ_(m) will vary with thespecific instance of the algorithm being run on the adiabatic quantumcomputer.

Other embodiments of the present invention can be constructed andoperated with a different estimate for the probability of a diabatictransition at step 404. For instance, in some embodiments the secondestimate of the Landau-Zener transition probability is computed. Thisprobability has the form: $\begin{matrix}{{P_{LZ}^{\prime} = {\mathbb{e}}^{{- 2}\quad\pi\quad k\quad\vartheta}};} & {{\vartheta = {\frac{1}{\hslash}\frac{g_{\min}^{2}}{E_{J}f_{P}}}},}\end{matrix}$where k is a constant that is about 1, E_(J) is the Josephson energy ofthe qubit (or maximum Josephson energy of the Josephson energies of aplurality of qubits), and f_(P) is frequency of oscillation of anadditional flux that is added to the superconducting qubit. The valuesfor the parameters g_(min), and E_(J) will vary with the specificinstance of the algorithm being run on the adiabatic quantum computer.

In many embodiments of the present invention, quantum systems foradiabatic quantum computation are designed such that the minimum of theenergy gap, the difference in the asymptotic slopes, and the rate ofchange of the adiabatic evolution parameter ensure that the probabilityof diabatic transition at step 404 is small, e.g. P_(LZ) is much smallerthan 1. In an embodiment of the present invention, P_(LZ) is 1×10⁻⁴ orless. In another embodiment of the present invention, P_(LZ) is 1×10⁻³or less. The probability of transition from the ground state of thequantum system to the first excited state of the quantum system isexponentially sensitive to the rate of change of the adiabatic evolutionand provides a lower bound to that rate. The probability of transitionfrom the ground state of the quantum system to the first excited stateof the quantum system also provides an upper limit on rate of change ofthe adiabatic evolution parameter. The duration of an adiabaticalgorithm, or process, should be less than the time it takes for aLandau-Zener transition to occur. If P_(LZ) is the probability peranticrossing, then the quantum system (e.g., quantum system 850 of FIG.8, which can be an individual qubit or a plurality of qubits) should bedesigned and operated such that the following inequality is satisfied:P_(LZ)×n_(A)<<1, where n_(A) is the number of anticrossings traversed intime T, or T<<(P_(LZ)×{dot over (γ)}×ρ_(A))⁻¹, where ρ_(A) is thedensity of anticrossings along the ground state of the energy spectra.The density of the anticrossings and crossings along the ground statecan be calculated by an approximate evaluation method. See, for exampleSection 5.3.3, which describes approximate evaluation methods, below.

In some embodiments of the present invention, the amount of timerequired to perform the readout process should be engineered so that theprobability that quantum system 850 will transition from the groundstate to the first excited state is small (e.g., a probability less thanone percent). For a readout process, the following should hold:P_(LZ)×m×r<<1, or τ<<f_(A)×(P_(LZ) ×m)⁻¹, where m is the number ofqubits, and r is the average number of cycles used to readout thequbits, f_(A) is the frequency of the cycles used to readout the qubits,and τ is the time for the readout of one qubit in a plurality of mqubits.

In an embodiment of the present invention this requisite smallprobability for transition P_(LZ) is dependent on the process performed.In the case of a readout process that applies additional flux to asuperconducting qubit and measures the state through a tank circuit,such as the embodiment described in detail below in conjunction withFIG. 8, the value of “small” is dependent on the number of cycles (r) ofthe waveform of the additional flux used. For example, in an embodimentof the present invention that reads out one qubit in one cycle, a smallP_(LZ) value is 1×10⁻² or less. In an embodiment of the presentinvention that reads out n qubits in r cycles, a small P_(LZ) value is(r×n)⁻¹×10⁻² or less. In an embodiment of the present invention thatreads out n qubits in r cycles, a small P_(LZ) value is (r×n)⁻¹10⁻⁴ orless. The cumulative probability of transition over the adiabaticprocess and subsequent readout cycles should be small and the systemdesigned and operated accordingly. In an embodiment of the presentinvention, a small cumulative probability of transition is about 1×10⁻²to about 1×10⁻⁶. In an embodiment of the present invention, a smallcumulative probability of transition is within ±0.05 or ±0.10 of thevalues given for P_(LZ) hereinabove.

5.3 Phase Qubit Embodiments

Section 5.3 describes quantum computing systems of the present inventionthat make use of phase qubits. Section 5.4, below, describes quantumcomputing systems of the present invention that make use of chargequbits.

5.3.1 Finding the Ground State of a Frustrated Ring Adiabatically Usinga Persistent Current Qubit Quantum System

FIG. 5A illustrates a first example of a quantum system 500 inaccordance with an embodiment of the present invention. As will bediscussed in detail in this section, in system 500, three coupled fluxqubits are dimensioned and configured in accordance with the presentinvention such that system 500 is capable of finding the ground state ofa frustrated quantum system using adiabatic computing methods.

5.3.1.1 General Description of the Persistent Current Qubit QuantumSystem Used in Exemplary Embodiments

Referring to FIG. 5A, each qubit 101 in quantum system 500 includes asuperconducting loop with three small-capacitance Josephson junctions inseries that enclose an applied magnetic flux fΦ_(o)(Φ_(o) is thesuperconducting flux quantum h/2e, where h is Planck's constant) and fis a number that can range from 0 (no applied flux) to 0.5 or greater.Each Josephson junction is denoted by and “×” within the correspondingsuperconducting loop. In each qubit 101, two of the Josephson junctionshave equal Josephson energy E_(J), whereas the coupling in the thirdjunction is αE_(J), with 0.5<α<1. Each qubit 101 has two stableclassical states with persistent circulating currents of opposite sign.For f=0.5, the energies of the two states are the same. The barrier forquantum tunneling between the states depends strongly on the value of α.Qubits 101 having the design illustrated in FIG. 5A has been proposed byMooij et al., 1999, Science 285, 1036, which is hereby incorporated byreference. The design and manufacture of such qubits is furtherdiscussed in Section 2.2.1, above.

The two stable states of a qubit 101 will have equal energy, meaningthat they will be degenerate, and will therefore support quantumtunneling between the two equal energy states (basis states) when theamount of flux trapped in the qubit is 0.5Φ_(o). The amount of fluxrequired to trap 0.5Φ_(o) in a qubit 101 is directly proportional to thearea of the qubit. Here, the area of a qubit is defined as the areaenclosed by the superconducting loop of the qubit. If the amount of fluxneeded to achieve a trapped flux of 0.5Φ_(o) in a first qubit 101 havingarea A₁ is B₁, then the amount of flux that is needed to trap 0.5Φ_(o)of flux in a second qubit having area A₂ is (A₂/A₁)B₁. Advantageously,in system 500, each qubit 101 has the same total surface area so that anexternal mechanism (e.g., a tank circuit) can cause each respectivequbit 101 in system 500 to trap 0.5Φ_(o) of flux at approximately orexactly the same time.

In preferred embodiments, the three persistent current qubits, 101-1,101-2, and 101-3 in structure 500 are inductively coupled to a tankcircuit (not fully shown in FIG. 5A). This tank circuit is comprised ofboth inductive and capacitive circuit elements. The tank circuit is usedto bias qubits 101 such that they each trap 0.5Φ_(o) of flux. In someembodiments of the present invention, the tank circuit has a highquality factor (e.g., Q>1000) and a low resonance frequency (e.g., afrequency less between 6 to 30 megahertz). The role of a tank circuit asa qubit control system is detailed in U.S. patent Publication2003/0224944 A1, entitled “Characterization and measurement ofsuperconducting structures,” as well as Il'ichev et al., 2004,“Radio-Frequency Method for Investigation of Quantum Properties ofSuperconducting Structures,” arXiv.org: cond-mat/0402559; and Il'ichevet al., 2003, “Continuous Monitoring of Rabi Oscillations in a JosephsonFlux Qubit,” Phys. Rev. Lett. 91, 097906, each of which is herebyincorporated by reference in its entirety. An inductive element of atank circuit is shown in FIG. 5A as element 502. In some embodiments,inductive element 502 is a pancake coil of superconducting material,such as niobium, with a nominal spacing of 1 micrometer between eachturn of the coil. The inductive and capacitive elements of the tankcircuit can be arranged in parallel or in series. For a parallelcircuit, a useful set of values for a small number of qubits is aninductance of about 50 nanohenries to about 250 nanohenries, acapacitance of about 50 picofarads to about 2000 picofarads, and aresonance frequency of about 10 megahertz to about 20 megahertz. In someembodiments, the resonance frequency f_(T) is determined by the formulaf_(T)=ω_(T)/2π=1/{square root}{square root over (L_(T)C_(T))} whereL_(T) is the inductance and C_(T) is the capacitance of the tankcircuit.

5.3.1.2 Selection of Persistent Current Qubit Device Parameters

In some embodiments of the present invention, qubit parameters arechosen to satisfy the requirements of the problem to be solved byadiabatic quantum computation and the restrictions of the qubits. Forinstance, in the case of MAXCUT, the couplings between qubits can bechosen so that they are greater than the energy of the tunneling term ofthe individual qubit Hamiltonians. Unfortunately, for a persistentcurrent qubit, the tunneling term is always non-zero and oftennon-variable. This presents a problem because, as was stated in Section5.1, in preferred embodiments, a problem Hamiltonian H_(P) is chosen forstep 403 of FIG. 4 that does not permit quantum tunneling to occur. Yet,in the case of persistent current qubits, a state that does not permitquantum tunneling cannot be found because in such qubits the tunnelingterm is always non-zero. However, in contrast to what was stated in thegeneral description of the adiabatic quantum processes of the presentinvention, it is not absolutely necessary to have the tunneling termabsent from the problem Hamiltonian H_(P) or the initial Hamiltonian H₀.In fact, some embodiments of the present inventions permit a non-zerotunneling term in H_(P) and or/or H₀ for any combination of thefollowing reasons: (i) the tunneling term leads to anticrossing usefulfor read out processes and (ii) the requirement of non-tunneling term isdeemed to be too strict in such embodiments. As to the latter point, itis sufficient to have a coupling term that is much stronger than thetunneling term in some embodiments of the present invention.

In one particular embodiment of a qubit 101 in quantum system 500 (FIG.5A), the critical current density of the Josephson junctions is 1000amperes per centimeter squared. The largest and strongest junction ofeach qubit 101 in system 500 has an area of about 450 nanometers by 200nanometers. The capacitance of the largest junction is 4.5 femtofaradsand the ratio between the Josephson energy and the charging energy isabout 100. The charging energy in such embodiments is e²/2C. The ratiobetween the weakest and strongest Josephson junction is 0.707:1. Thetunneling energies of qubits 101 in this embodiment are each about 0.064Kelvin. The persistent current is 570 nanoanperes. For this value of thepersistent current and for inter-qubit mutual inductances taken from thedesign of FIG. 5A, the coupling energies between the qubits areJ_(1,2)=0.246 Kelvin, J_(2,3)=0.122 Kelvin, and J_(1,3)=0.059 Kelvin.All these parameters are within reach of current fabrication technology.The specific values for this exemplary embodiment are provided by way ofexample only and do not impose any limitation on other embodiments ofsystem 500.

Various embodiments of the present invention provide different valuesfor the persistent current that circulates in persistent current qubits101. These persistent currents range from about 100 nanoamperes to about2 milliamperes. The persistent current values change the slope of theasymptotes at anticrossing 915 (FIG. 9A). Specifically, the qubit biasis equal to πI_(P)(2Φ_(E)/Φ₀−1), where I_(P) is the persistent currentvalue and the slope of the asymptote (e.g., asymptote 914 and/or 916) isproportional to the qubit bias, for large bias, when such bias is about10 times the tunneling energy. In some embodiments of the presentinvention, qubits 101 have a critical current density of about 100 A/cm²to about 2000 A/cm². In some embodiments of the present invention,qubits 101 have a critical current that is less than about 200nanoamperes, less than about 400 nanoamperes, less than about 500nanoamperes or less than about 600 nanoamperes. In some embodiments ofthe present invention, the term “about” in relation to critical currentmeans a variance of ±5%, ±10%, ±20%, and ±50% of the stated value.

5.3.1.3 Algorithm Used to Solve the Computational Problem

Step 401 (Preparation). An overview of an apparatus 500 used to solve acomputational problem in accordance of the present invention has beendetailed in Section 5.3.1.1. In this section, the general adiabaticquantum computing process set forth in FIG. 4 is described. In the firststep, preparation, the problem to be solved and the system that will beused to solve the problem is described. Here, the problem to be solvedis finding or confirming the ground state of a three node frustratedring. System 500 is used to solve this problem. Entanglement of one ormore qubits 101 is achieved by the inductive coupling of flux trapped ineach qubit 101. The strength of this type of coupling between twoabutting qubits is, in part, a function of the common surface areabetween the two qubits. Increased common surface area between abuttingqubits leads to increased inductive coupling between the two abuttingqubits.

In accordance with the present invention, the problem of determining theground state of the three node frustrated ring is encoded into system500 by customizing the coupling constants between neighboring qubits 101using two variables: (i) the distance between the qubits and (ii) theamount of surface area common to such qubits. The lengths and widths ofqubits 101, as well as the spatial separation between such qubits, isadjusted to customize inter-qubit inductive coupling strengths in such away that these coupling strengths correspond to a computational problemto be solved (e.g., the ground state of a three member frustrated ring).In preferred embodiments, qubit 101 length and width choices are subjectto the constraint that each qubit 101 have the same or approximately thesame total surface area so that the qubits can be adjusted to a statewhere they each trap half a flux quantum at the same time.

As shown in FIG. 5A, the configuration of qubits 101 represents a ringwith inherent frustration. The frustrated ring is denoted by the dashedtriangle in FIG. 5A through qubits 101. Each two qubits of the set{101-1, 101-2, 101-3} has a coupling that favors antiferromagneticalignment, i.e. with flux aligned down and up, in the same way that barmagnets align NS with SN. Because of the presence of the odd third qubitand the asymmetry that results from the odd third qubit, system 500 doesnot permit such an alignment of coupling. This causes system 500 to befrustrated. In general, a ring-like configuration of an odd number ofqubits will result in a frustrated system. Referring to FIG. 5A, in anembodiment of the invention, the area of each qubit 101 is approximatelyequal but the heights (y dimension) and widths (x dimension) areunequal. In one specific example, persistent current qubits, such asqubits 101, with an area of about 80 micrometers squared (e.g. height ofabout 9 micrometers and width of about 9 micrometers, height of 4micrometers and width of 40 micrometers, etc.) are arranged with twocongruent qubits paired lengthwise (e.g., 101-1 and 101-2) and a thirdnon-congruent qubit (e.g., 101-3) laid transverse and abutting the endof the pair, as shown in FIG. 5A.

All three qubits 101-1, 101-2, and 101-3, have the same area subject tomanufacturing tolerances. In some embodiments of the present invention,such tolerance allows for a ±25% deviation from the mean qubit surfacearea, in other embodiments such tolerance is ±15%, ±5%, ±2%, or lessthan ±1% deviation from the mean qubit surface area. Qubits 101-1,101-2, and 101-3 are coupled to each other asymmetrically. In otherwords, the total surface area common to qubit 101-3 and 101-1 (or 101-2)is less than the total surface area common to qubit 101-1 and 101-2.

Embodiments of the present invention, such as those that include asystem like 500, have a Hamiltonian given by:$H = {{\sum\limits_{i = 1}^{N}\left\lbrack {{ɛ_{i}\quad\sigma_{i}^{Z}} + {\Delta_{i}\sigma_{i}^{X}}} \right\rbrack} + {\sum\limits_{i = 1}^{N}{\sum\limits_{j > i}^{N}{J_{ij}\quad{\sigma_{i}^{Z} \otimes \sigma_{j}^{Z}}}}}}$where N is the number of qubits. The quantity Δ_(i) is the tunnelingrate, or energy, of the i^(th) qubit expressed in units of frequency, orenergy. These unit scales differ by a factor of h or Plank's constant.The quantity ε_(i) is the amount of bias applied to the qubit, orequivalently, the amount of flux in the loops of the qubits. Thequantity J_(ij) is the strength of the coupling between the i^(th) andthe j^(th) qubit. The coupling energy is a function of the mutualinductance between qubits J_(ij)=M_(ij)I_(i)I_(j). In order to encodethe three node frustrated ring problem, the three coupling energies arearranged such that J₁₂>>J₁₃≈J₂₃.

In some embodiments of the present invention, the coupling strengthbetween any two abutting qubits is a product of the mutual inductanceand the currents in the coupled qubits. The mutual inductance is afunction of common surface area and distance. The greater the commonsurface area the stronger the mutual inductance. The greater thedistance the less the mutual inductance. The current in each qubit is afunction of the Josephson energy of the qubit, E_(J). E_(J) depends onthe type of Josephson junctions interrupting the superconducting loopwithin the qubit. Inductance calculations can be performed usingprograms such as FASTHENRY, a numeric three-dimensional inductancecalculation program, is freely distributed by the Research Laboratory ofElectronics of the Massachusetts Institute of Technology, Cambridge,Mass. See, Kamon, et al., 1994, “FASTHENRY: A Multipole-Accelerated 3-DInductance Extraction Program,” IEEE Trans. on Microwave Theory andTechniques, 42, pp. 1750-1758, which is hereby incorporated by referencein its entirety. Alternatively, mutual inductance can be calculatedanalytically using techniques known in the art. See Grover, InductanceCalculations: Working Formulas and Tables, Dover Publications, Inc., NewYork, 1946, which is hereby incorporated by reference in its entirety.

Step 403 (Initialization of system 500 to H_(o)). Qubits 101 of system500 of FIG. 5A interact with their environment through a magnetic fieldthat is applied perpendicular to the plane of FIG. 5A by causing acurrent to flow through coil 502 of the tank circuit. In step 403,system 500 is set to an initial state characterized by the HamiltonianH_(o) by applying such an external magnetic field. This externallyapplied magnetic field creates an interaction defined by the followingHamiltonian: $H_{0} = {Q{\sum\limits_{i = 1}^{N}\sigma_{i}^{Z}}}$where Q represents the strength of the external magnetic field. In someembodiments, this external magnetic field has a strength such that ε>>Δor ε>>J. That is, the magnetic field is on an energy scale that is largerelative to other terms in the Hamiltonian. As an example, for a qubithaving the dimensions of 50 to 100 micrometers squared, the magneticfield is between about 10⁻⁸ teslas and about 10⁻⁶ teslas. The energy ofa magnetic field B in a persistent current qubit of area S, in terms ofMKS units is ${\,_{\frac{1}{2\mu_{o}}}S} \cdot B^{2}$where μ₀ is 4π×10⁻⁷ Wb/(A·m) (webers per ampere meter). The magneticfield also controls the bias ε applied to each qubit. In step 403, thelarge external magnetic field is used to initialize system 500 to thestarting state characterized by the Hamiltonian H₀. System 500 is in theground state |000> of H₀ when the flux that is trapped in each qubit 101is aligned with the external magnetic field.

Once system 500 is in the ground state |000> of the starting state H₀,it can be used to solve the computational problem hard-coded into thesystem through the engineered inter-qubit coupling constants. Toaccomplish this, the applied magnetic field is removed at a rate that issufficiently slow to cause the system to change adiabatically. At anygiven instant during the removal of the externally appliedinitialization magnetic field, the state of system 500 is described bythe instantaneous Hamiltonian:$H = {{\sum\limits_{i = 1}^{N}\left\lbrack {{ɛ_{i}\sigma_{i}^{Z}} + {\Delta_{i}\sigma_{i}^{X}}} \right\rbrack} + {\sum\limits_{i = 1}^{N}{\sum\limits_{j > i}^{N}{J_{ij}{\sigma_{i}^{Z} \otimes \sigma_{j}^{Z}}}}}}$where Δ is a tunneling term that is a function of the bias ε applied bycoil 502. As described above, it is preferred that the bias ε applied bycoil 502 be such that a half flux quantum (0.5Φ_(o)) be trapped in eachqubit 101 so that the two stable states of each qubit have the sameenergy (are degenerate) to achieve optimal quantum tunneling.Furthermore, it is preferred that each qubit have the same total surfacearea so that quantum tunneling in each qubit starts and stops at thesame time. In less preferred embodiments, qubits 101 are biased by coil502 such that the flux trapped in each qubit is an odd multiple of 0.5Φ₀(e.g., 1.5, 2.5, 3.5, etc., or in other words, N·0.5Φ₀, where N is 1, 2,3, . . . ) since the bistable states of the qubits 101 will have equalenergy (will be degenerate) in these situations as well. However, suchhigher amounts of trapped flux can have undesirable side effects on theproperties of the superconducting current in the qubits. For instance,large amounts of trapped flux in qubits 101 may quench thesuperconducting current altogether that flows through thesuperconducting loop of each said loop. In some embodiments, the qubitsare biased by coil 502 such that the amount of flux trapped in eachqubit is ζ·N·0.5Φ₀ where ζ is between 0.7 and 1.3, between 0.8 and 1.2,or between 0.9 and 1.1 and where N is a positive integer. However,qubits 101 tunnel best when the bias in each corresponds to half-fluxquantum of flux.

Step 405 (reaching the problem state H_(P)). As the externalperpendicularly applied field is adiabatically turned off, the problemHamiltonian H_(P) is arrived at:$H_{P} = {\sum\limits_{i = 1}^{N}{\sum\limits_{j > i}^{N}{J_{ij}{\sigma_{i}^{Z} \otimes \sigma_{j}^{Z}}}}}$Note that, unlike the instantaneous Hamiltonian, the problem Hamiltoniandoes not include the tunneling term Δ. Thus, when system 500 reaches thefinal problem state, the quantum states of each respective qubit 101 canno longer tunnel. Typically, the final state is one in which no externalmagnetic field is applied to qubits 101 and, consequently, qubits nolonger trap flux.

Step 407 (Measurement). In step 407, the quantum system is measured. Inthe problem addressed in the present system, there eight possiblesolutions {000, 001, 010, 100, 011, 110, 101, and 111}. Measurementinvolves determining which of the eight solutions was adopted by system500. An advantage of the present invention is that this solution willrepresent the actual solution of the quantum system.

In an embodiment of the present invention, the result of the adiabaticquantum computation is determined using individual qubit magnetometers.Referring to FIG. 5E, a device 517 for detecting the state of apersistent current qubit is placed proximate to each qubit 101. Thestate of each qubit 101 can then be read out to determine the groundstate of H_(P). In an embodiment of the present invention, the device517 for detecting the state of the qubit is a DC-SQUID magnetometer, amagnetic force microscope, or a tank circuit dedicated to a single qubit101. The read out of each qubit 101 creates an image of the ground stateof H_(P). In embodiments of the present invention the device for readingout the state of the qubit encloses the qubit. For example a DC-SQUIDmagnetometer like 517-7, 517-8, and 517-9 in FIG. 5E, may enclose apersistent current qubit, e.g. 101-7, 101-8, and 101-9, to increase thereadout fidelity of the magnetometer.

In an embodiment of the present invention, the result of the adiabaticquantum computation is determined using individual qubit magnetometerslaid adjacent to respective qubits in the quantum system. Referring toFIG. 5F, devices 527-1, 527-2, and 527-3 are for detecting the state ofa persistent current qubit and are respectively placed besidecorresponding qubits 501-1, 501-2, and 501-3. The state of each qubit501 can then be read out to determine the ground state of H_(P). In anembodiment of the present invention, the device for detecting the stateof a qubit 501 is a DC-SQUID magnetometer, dedicated to one qubit.

In an embodiment of the present invention, the qubits 501 used in theadiabatic quantum computation are coupled by Josephson junctions.Referring to FIG. 5F, qubits 501-1, 501-2, and 501-3 are coupled to eachother by Josephson junctions 533. In particular, Josephson junction533-1 couples qubit 501-1 to qubit 501-2, Josephson junction 533-2couples qubit 501-2 to qubit 501-3, junction 533-3 couples qubit 501-3to qubit 501-1. The sign of the coupling is positive foranti-ferromagnetic coupling, the same as inductive coupling. The energy(strength) of the coupling between persistent current qubits 501 thatare coupled by Josephson junctions can be about 1 kelvin. In contrast,the energy of the coupling between persistent current qubits 501 thatare inductively coupled is about 10 milikelvin. In an embodiment of thepresent invention, the term “about” in relation to coupling energies,such as Josephson junction mediated and inducted coupling betweenpersistent current qubits, is defined as a maximum variance of ±10%,±50%, ±100%, or ±500% of the energy stated. The coupling energy betweentwo qubits is proportional 2I²/E where I is the current circulating thequbits, and E is the Josephson energy of the coupling Josephsonjunction. See, Grajcar et al., 2005, arXiv.org: cond-mat/0501085, whichis hereby incorporated by reference in its entirety.

In an embodiment of the present invention, qubits 501 used in theadiabatic quantum computation have tunable tunneling energies. Referringto FIG. 5F, in such an embodiments, qubits 501-1, 501-2, and 501-3include split Josephson junctions 528. Other qubits, described hereincan make use of a split junction. Split Josephson junction 528-1 isincluded in qubit 501-1 in the embodiments illustrated in FIG. 5F.Further, the split Josephson junction 528-2 is included in qubit 501-2,the split Josephson junction 528-3 is included in qubit 501-3. Eachsplit junction 528 illustrated in FIG. 5F includes two Josephsonjunctions and a superconducting loop in a DC-SQUID geometry. The energyof the Josephson junction E_(J), of the qubit 501, which is correlatedwith the tunnelling energy of the qubit, is controlled by an externalmagnetic field supplied by the loop in the corresponding split Josephsonjunction 528. The Josephson energy of split 528 can be tuned from abouttwice the Josephson energy of the constituent Josephson junctions toabout zero. In mathematical terms,$E_{J} = {2E_{J}^{0}{{\cos\left( \frac{\pi\quad\Phi_{X}}{\Phi_{0}} \right)}}}$where Φ_(X) is the external flux applied to the split Josephsonjunction, and E_(j) ⁰ is the Josephson energy of one of the Josephsonjunctions in the split junction. When the magnetic flux through splitjunction 528 is one half a flux quantum the tunneling energy for thecorresponding qubit 501 is zero. The magnetic flux in the splitJosephson junctions 528 may be applied by a global magnetic field.

In an embodiment of the present invention the tunneling of qubits 501 issuppressed by applying a flux of one half a flux quantum to the splitJosephson junction 528 of one or more qubits 501. In an embodiment ofthe present invention the split junction loop is orientatedperpendicular to the plane of the corresponding (adjacent) qubit 501such that flux applied to it is transverse in the field of the qubits.

Referring to FIG. 5, in an embodiment of the present invention, thequbits used in the adiabatic quantum computation have both ferromagneticand antiferromagnetic couplings. The plurality of qubits 555 in FIG. 5Gare coupled both by ferromagnetic and antiferromagnetic couplings.Specifically, the coupling between qubits 511-1 and 511-3, as well asbetween qubits 511-2 and 511-4, are ferromagnetic while all othercouplings are antiferromagnetic (e.g., between qubits 511-1 and 511-2).The ferromagnetic coupling is induced by crossovers 548-1 and 548-2.

Referring to FIG. 5A, in some embodiments of the present invention, thestate of each qubit 101 of system 500 is not individually read out.Rather, such embodiments make use of the profile of the energy leveldiagram of system 500, as a whole, over a range of probing biasingcurrents. In such embodiments, when system 500 reaches H_(P) the systemis probed with a range of biasing currents using, for example, a tankcircuit that includes coil 502. The overall energy level of system 500over the range of biasing currents applied during the measurement step407 define an energy level profile for the system 500 that ischaracteristic of the state of system 500. Furthermore, as discussed inmore detail below, system 500 is designed in some embodiments such thateach of the possible eight states that the system could adopt has aunique calculated energy profile (e.g., unique number of inflectionpoints, unique curvature). Therefore, in such embodiments, measurementof H_(P) can be accomplished by computing the energy profile of system500 with respect to a range of biasing currents. In such embodiments,the state of system 500 (e.g., 001, 010, 100, etc.) is determined bycorrelating the calculated characteristics (e.g., slope, number ofinflection point, etc.) of the measured energy profile to thecharacteristics of the energy profile calculated for each of thepossible solutions (e.g., the characteristics of a calculated energyprofile for 001, the characteristics of a calculated energy profile for010, and so forth). When designed in accordance with the presentinvention, there will only be one calculated energy profile that matchesthe measured energy profile and this will be the solution to the problemof finding the ground state of a frustrated ring.

In other embodiments of the present invention, the state of each qubit101 of system 500 is not individually read out. Rather, such embodimentsmake use of the profile of the energy level diagram of system 500 as awhole over a range of probing biasing currents. In such embodiments,when system 500 reaches H_(P) the system is probed with a range ofbiasing currents using, for example, a tank circuit that includes coil502 (FIG. 5A). The overall energy level of system 500 over the range ofbiasing currents applied during the measurement step 407 define anenergy level profile for the system 500 that is characteristic of thestate that the system is in. Furthermore, as discussed in more detailbelow, system 500 can be designed such that each of the possible eightstates that the system could have is characterized by a uniquecalculated energy profile (e.g., unique number of inflection points,unique curvature). Therefore, in such embodiments, measurement of H_(P)could involve computing the energy profile of system 500 with respect toa range of biasing currents and then determining the state of system 500(e.g., 001, 010, 100, etc.) by correlating the calculatedcharacteristics (e.g., slope, number of inflection point, etc.) of themeasured energy profile to the characteristics of the energy profilecalculated for each of the possible solutions (e.g., the characteristicsof a calculated energy profile for 001, the characteristics of acalculated energy profile for 010, and so forth). When designed inaccordance with this embodiment of the present invention, there willonly be one calculated energy profile that matches the measured energyprofile and this will be the solution to the problem of finding theground state of the modeled frustrated ring.

As described above, for many embodiments of the present invention, theenergy levels of the system for adiabatic quantum computing H_(P) acrossa range of probing biasing values are differentiable from each other.FIG. 7B illustrates an example of this. Consider a system 500 in whichtwo of the three inter-qubit coupling constants are equal toδ(J₁₃=J₂₃=δ) and the third is equal to three times that amount J₁₂=3·δ.Further, the tunneling rates of the qubits are about equal and, in factare about equal to the small coupling value δ(e.g., Δ₁=1.1·δ; Δ₂=δ;Δ₃=0.9·δ) where ·δ is in normalized units. In an embodiment of thepresent invention, δ can be a value in the range of about 100 megahertzto about 20 gigahertz. When system 500 is configured with thesetunneling and coupling values then the curvature of the two lowestenergy levels will be different from each other and therefore can bedistinguished by the impedance techniques described above. In theseimpedance techniques, the final bias current applied by the tank circuitin system 500 is not fixed. Rather, it is adjusted to produce the energylevels of FIG. 7B. Indeed, FIG. 7B is plotted as a function of energy Eversus bias ε, for an example where the areas encompassed by each of thequbits 101 are equal (e.g., system 500 of FIG. 5A). The bias ε is inunits of energy and the scale of the horizontal axis of FIG. 7B has isin units equal to Δ. The bias is the same for each qubit, because thearea is same for each qubit. By contrast, FIG. 7A shows the energylevels for various instantaneous times during the adiabatic change froma state H₀ to a state H_(P).

Considering FIG. 7B, embodiments of the present invention use existingtechniques for differentiating energy levels by shape, where the shapeor curvature of a first energy level as a function of bias ε can bedifferentiated from other energy levels. See, for example, U.S. patentPublication 2003/0224944 A1, entitled “Characterization and measurementof superconducting structures,” which is hereby incorporated byreference in its entirety. The impedance readout technique can be usedto readout the system 500 to determine the states of the qubits in thefrustrated arrangement. The unique curvature for each possible energylevel (solution) does not significantly change if J₁₃ is not exactlyequal to J₂₃.

In an embodiment of the present invention, the system is read out bydifferentiating various energy levels (solutions) to identify the groundstate. In an example of the persistent current qubit, the dephasing rateis currently recorded as being 2.5 microseconds or less. Once the systemis in a state of H_(P) the state, e.g. 551 or 550 can be determined bylocally probing the energy level structure by low frequency applicationsof a biasing magnetic field.

The states of energy level diagram like that of FIG. 7B can bedifferentiated through the curvatures, and number of inflection pointsof the respective energy levels. Two energy levels may have a differentsign on the curvature that allows them to be distinguished. Two energylevels may have the same signs but have different magnitudes of thecurvature. Two energy levels may have a different number of inflectionpoints. All of these generate different response voltages in the tankcircuit. For instance, an energy level with two inflection points willhave a voltage response for each inflection. The sign of the voltageresponse is correlated with the sign of the curvature.

Knowledge of the initial energy level and the corresponding voltageresponse can allow the ground state to be determined provided that asampling of some of the lowest energy levels voltage response has beenmade. Accordingly, in an embodiment of the present invention, the systemis not initialized in the ground state of the initial Hamiltonian.Rather the system is initialized in an excited state of the system suchas 551. Then the interpolation between initial and final Hamiltoniansoccurs at a rate that is adiabatic but is faster than the relaxationrate out of state 551. In one example of a persistent current qubit, therelaxation rate is about 1 to about 10 microseconds. The ground statewith have a greater curvature and the lowest number inflection points ofthe energy levels of the system.

5.3.2 Observation and Readout with a Superconducting Tank Circuit

FIG. 8 illustrates a generic embodiment tank circuit and associateddevices used to control superconducting qubits in accordance with someembodiments of the present invention. FIG. 8 includes a superconductingqubit or qubits 850 (quantum system), a tank circuit 810, and anexcitation device 820. System 800 is arranged geometrically such thatquantum system 850 has a mutual inductance M′ with excitation device820, and a mutual inductance M with tank circuit 810. In an embodimentof the present invention, tank circuit 810 includes an inductance L_(T)(805), capacitance C_(T) (806), and a frequency dependent impedanceZ_(T)(f) (804), where f is the response frequency of the tank. Tankcircuit 810 can further include one or more Josephson junctions 803,which can be biased to provide a tunable inductance. Tank circuit 810has a resonant frequency f_(o) that depends on the specific values ofits characteristics, such as the L_(T), C_(T), and Z(f) components. Anembodiment of the present invention can make use of multiple inductors,capacitors, Josephson junctions, or impedance sources but, without lossof generality, the generalized circuit depiction illustrated in FIG. 8can be used to describe tank circuit 810. In other words, inductance 805represents one or more inductors in series or parallel, Josephsonjunction 803 is one Josephson junction or more than one Josephsonjunction in series or parallel, capacitor 806 represents a singlecapacitor or more than one capacitor in series or parallel, andimpedance 804 represents one or more impedance sources in series orparallel. Furthermore, although inductance 805, Josephson junctions 803,capacitance 806 and impedance 804 is shown in series, tank circuits 810are not limited to such a configuration. These electrical components canbe arranged in any configuration, including series and parallel,provided that circuit 810 maintains tank circuit functionality as isknown in the art.

In operation, when an external signal, such as a magnetic flux in aninductively coupled device, is applied through tank circuit 810, aresonance is induced. This resonance affects the magnitude of impedance804. In particular, impedance 804 is affected by the direction of themagnetic flux in devices inductively coupled to it, such as asuperconducting qubit (e.g., quantum system 850). Because of the statedependency of impedance 804, the tank voltage or current response islikewise state dependent. Therefore, the tank voltage or currentresponse of tank circuit 810 can be used to measure the direction of themagnetic flux in devices inductively coupled to the tank circuit. In anembodiment of the present invention, tank circuit 810 is inductivelycoupled to a superconducting qubit (an example of quantum system 850)having quantum states such that the resonant frequency of the tankcircuit is correlated with the state of the superconducting qubit. Inoperation, characteristics of the superconducting qubit can be probed bytank circuit 810 by applying signals through tank circuit 810 andmeasuring the response of the tank circuit. The characteristics of tankcircuit 810 can be observed by amplifier 809, for example. See, forexample, U.S. patent Publication 2003/0224944 A1 , entitled“Characterization and measurement of superconducting structures,” whichis hereby incorporated by reference in its entirety.

Some embodiments of the present invention make use of the high electronmobility field-effect transistors (HEMT) and their improvedpseudomorphic variants (PHEMT) to measure the signal from tank circuit810. Such embodiments can make use of multistage amplifiers 809, such astwo, three, or four transistors. Some embodiments of the presentinvention use the commercially available transistor ATF-35143 fromAgilent Technologies, Inc.(Palo Alto, Calif.) as amplifier 809. In someembodiments, amplifier 809 comprises a three transistor amplifier havinga noise temperature of about 100 millikelvin. The power consumption ofthe transistors is decreased by reducing both the transistors' drainvoltage and the drain current to about two percent of the averageratings for the drain source voltage, and about 0.3% of the currentrating. This reduces the power dissipation of the first transistor toabout 20 milliwatts. In some embodiments, all active resistances in theamplifier's signal channel are replaced by inductors and third stage isused to match the amplifier output with cable impedance. In someembodiments, amplifier 809 is assembled on a printed board. In aspecific example, amplifier 809 is a cold amplifier, built as describedabove, and is thermally connected to a helium-3 pot of a dilutionrefrigerator that houses quantum system 850.

In one embodiment of the present invention, the response voltage of thetank circuit is amplified by a cold amplifier thermally coupled to thehelium-3 pot of the dilution refrigerator. The response signal is oncemore amplified by a room temperature amplifier and detected by an rflock-in voltmeter. The rf lock-in voltmeter can be used to average theresponse of the tank circuit over many cycles of the response of thetank. For more information on such an amplifier arrangement, seeOukhanski et al., 2003, Rev. Sci. Instr. 72, 1145, which is herebyincorporated by reference in its entirety.

Some embodiments of the present invention make use of tank circuit 810to read out the state of quantum system 850. By making use of aradio-frequency technique, called the impedance measurement technique(IMT), the main parameters of the circuit described by quantum system850 can be reconstructed. With the IMT, a qubit in quantum system 850 tobe read out is coupled through a mutual inductance M to a tank circuit810 (FIG. 8) with known inductance L_(T), capacitance C_(T), and qualityfactor Q. In a specific embodiment of the present invention, tankcircuit 810 has M that is about 180 picohenries, L_(T) that is about 118nanohenries, C_(T) that is about 470 picohenries, and a quality factor Qthat is about 2400. In another specific embodiment of the presentinvention, tank circuit 810 has an M that is about 55 picohenries, anL_(T) that is about 85 nanohenries, C_(T) that is about 470 picohenries,and a quality factor Q that is about 436. In still another embodiment ofthe present invention, tank circuit 810 has a quality factor Q that isabout 1500. In still another embodiment of the present invention, tankcircuit 810 has a quality factor Q that is between 1400 and 1600.

Tank circuit 810 is driven by a direct bias current I_(DC) and analternating current I_(RF) of a frequency ω close to the resonancefrequency of the tank circuit ω₀. The total externally applied magneticflux to the qubit, Φ_(E), is therefore determined by the tank circuitwhere Φ_(E)=Φ_(DC)+ΦO_(RF). Here, Φ_(DC)=Φ_(A)+f(t)Φ₀.

In one embodiment of the present invention, the direct bias currentI_(DC) and an alternating current I_(RF) in the tank circuit aresupplied along filtered lines by commercially available current sources.An example of a DC-current source for supplying the direct bias currentI_(DC) is the Agilent 33220A. An example of an AC-current source forsupplying the alternating current I_(RF) is the Agilent E8247C microwavegenerator. Both devices are available from Agilent Technologies, Inc.(Palo Alto, Calif.).

The terms Φ_(A) and f(t)Φ₀ are constant or are slowly varying and aregenerated by the direct bias current I_(DC). The magnitude of Φ_(A) isdiscussed below, f(t) has a value between 0 and 1 inclusive, and Φ₀ isone flux quantum (2.07×10⁻¹⁵ webers (or volt seconds)). Further, Φ_(RF)is a small amplitude oscillating rf flux generated with ac currentI_(RF), having a magnitude of about 10⁻⁵Φ₀ to about 10⁻¹Φ₀. In anembodiment of the present invention, the Φ_(RF) amplitude is about10⁻³Φ₀. In an embodiment of the present invention, the term “about” inrelation to small magnetic fluxes, such as Φ_(RF), means a variance of±10%, ±50%, ±100%, and ±500% of the magnetic flux.

If the amplitude of Φ_(RF) is small, meaning that it has a negligiblevalue relative to Φ₀(Φ_(RF)<<Φ₀), then the approximate equalityΦ_(E)≅Φ_(DC) is valid. In an embodiment of the present invention f(t),Φ_(A), and Φ_(RF) are varied on progressively shorter time scalesbecause f(t) and Φ_(A) are the qubit bias, and Φ_(RF) is a smallamplitude fast function that is used to determine the state of quantumsystem 850 (e.g., the state of a qubit in quantum system 850). Bymonitoring the effective impedance of tank circuit 810 (FIG. 8) as afunction of the externally applied magnetic flux Φ_(E), the property ofa superconducting qubit in the quantum system 850 sharing a mutualinductance M with tank circuit 810, as depicted in FIG. 8, can bedetermined. This is equivalent to monitoring the magnetic susceptibilityof the superconducting qubit. Although FIG. 8 depicts quantum system 850as being outside tank circuit 810, those of skill in the art willappreciate that in some embodiments, quantum system 850 is containedwithin tank circuit 810.

The imaginary part of the total impedance of tank circuit 810, expressedthrough the phase angle χ between driving current I_(RF) and tankvoltage, can be expressed as${\tan\quad{\chi\left( \Phi_{DC} \right)}} = \frac{k^{2}Q\quad\beta\quad{F^{\prime}\left\lbrack {\Phi\left( \Phi_{DC} \right)} \right\rbrack}}{1 + {\beta\quad{F^{\prime}\left\lbrack {\Phi\left( \Phi_{DC} \right)} \right\rbrack}}}$at ω=ω₀. Here, F′(Φ(Φ_(DC)))=I′(Φ(Φ_(DC))/I) _(C) is the derivative ofthe normalized supercurrent in the qubit with respect to the totalmagnetic flux, Φ=Φ_(E)−β×F(Φ(Φ_(DC))) in the superconducting qubit. Theparameter β=2πL I_(c)/Φ₀ is the normalized inductance of thesuperconducting qubit, having inductance L, and critical current I_(C).

5.3.2.1 Observation and Readout by Traversing Crossings andAnticrossings

This section describes techniques for reading out the state of a quantumsystem 850. In some embodiments, this is accomplished by reading out thestate of each qubit within quantum system 850 on an individual basis.Individualized readout of qubits in a quantum system 850 can beaccomplished, for example, by making use of individual bias wires orindividual excitation devices 820 for each qubit within quantum system850 as described in this section and as illustrated in FIG. 5B, in thecase of individual bias wires. In some embodiments, the entire processdescribed in FIG. 4 is repeated and, during each repetition a differentqubit in the quantum system 850 is measured. In some embodiments,described in Section 5.3.2.6, it is not necessary to repeat the processdescribed in FIG. 4 for each qubit in quantum system 850.

FIG. 9 illustrates sections of energy level diagrams with energy levelcrossing and anticrossing. FIGS. 9A and 9B are useful for describing howthe readout of superconducting qubits in quantum system 850 works, andhow one performs such a readout. FIGS. 9A and 9B show energy levels fora qubit as a function of external flux Φ applied on the qubit. In someembodiments, aspects of system 800 from FIG. 8 can be represented byFIGS. 9A and 9B. For example, superconducting qubit 850 of FIG. 8 canhave an energy level crossing or anticrossing depicted in FIG. 9 thatcan be probed by tank circuit 810.

FIG. 9A illustrates an energy level diagram with an anticrossing. Ananticrossing arises between energy levels of a qubit when there is atunneling term or, more generally, a transition term between the levels.Energy levels 909 (ground state) and 919 (excited state) have ahyperbolic shape as a function of applied magnetic flux, with ananticrossing within box 915, and approach asymptotes 914 and 916. Thevalue of Δ_(m) for the energy level diagram is the difference in theslopes of the pair of asymptotes 914 and 916 near the anticrossing.Lines 914 and 915 of FIG. 9A are an example of a set of asymptotes nearan anticrossing. The superconducting qubit, part of a quantum system850, and described by the energy diagram of FIG. 9A, has computationalbasis states |0> and |1>. For a description of such basis states, seeSection 2.1, above. In an embodiment of the present invention, where thesuperconducting qubit is a three-Josephson junction qubit, the |0> and|1> computational basis states correspond to right and left circulatingsupercurrents (102-0 and 102-1 of FIG. 1A). The |0> and |1>computational basis states are illustrated in FIG. 9A. In the groundstate 909 of the qubit represented by FIG. 9A, the |1> basis statecorresponds to region 910 to the left of degeneracy point 913 and the|1> basis state corresponds to region 911 on the right of the degeneracypoint. In the excited state 919 of the qubit represented by FIG. 9A, the|0> basis state is on the right of degeneracy point 923 in region 920while the |1> basis state corresponds to region 921 on the left ofdegeneracy point 923.

In accordance with an embodiment of the present invention, a method forreading out the state of a superconducting qubit within quantum system850 involves using tank circuit 810 to apply a range of fluxes to thesuperconducting qubit over a period of time and detecting a change inthe properties of the tank circuit coupled to the superconductingcircuit during the sweep. In the case where the qubit is asuperconducting flux qubit, a superconducting loop with low inductance Linterrupted by three Josephson junctions, the flux applied during thesweep can range from, for example, 0.49Φ₀ to 0.51Φ₀. In otherembodiments, the flux applied during the sweep can range between 0.3Φ₀and 0.70Φ₀. In still other embodiments, the flux applied during thesweep can range across several multiples of Φ₀ (e.g., between 0.3Φ₀ and5Φ₀ or more). In preferred embodiments of the present invention, theflux sweep is performed adiabatically to ensure that the qubit remainsin its ground state during the transition. Under this adiabatic sweep,tank circuit 810 (FIG. 8) detects when the quantum state of the qubitcrosses the anticrossing and hence determines the quantum state of thequbit at the end of step 405 of FIG. 4. For example, referring to FIG.9A, if the qubit is in ground state |0> at the end of step 405 on theleft of the energy level anticrossing 915 and the flux is adiabaticallyincreased during step 407, then the tank circuit will detectanticrossing 915. On the other hand, if the qubit is in ground state |1>at the end of step 405 on the right of anticrossing 915, then when theflux is increased the qubit will not cross anticrossing 915 andtherefore no such anticrossing will be detected. In this way, providedthat the qubit is maintained in the ground state, the applied sweep offluxes can be used to determine whether the qubit was in the |0> or the|1> basis state prior to readout.

The readout of a superconducting qubit involves sweeping the fluxapplied to the superconducting qubit by an amount and in a directiondesigned to detect the anticrossing 915. This method makes use of thefact that a tank circuit can detect the anticrossing 915 by the changein the curvature of the ground state energy level 909 at anticrossing915. In other words, the inventive method measures the magneticsusceptibility of the qubit. See, for example, U.S. patent Publication2003/0224944 A1, entitled “Characterization and measurement ofsuperconducting structures,” which is hereby incorporated by referencein its entirety. In some embodiments, the measurable quantity is a dipin the voltage across tank circuit 810, or a change in the phase of tankcircuit 810, or both. If a dip in the voltage results from the readoutoperation then the state of the qubit within quantum system 850transitioned through the anticrossing. If no dip is observed, the qubitdid not transition through the anticrossing during measurement sweep407. These state-dependent observables are used to perform the readoutoperation.

In some embodiments, when the superconducting qubit is in the |0> groundstate, e.g. in region 910, and the flux applied to the superconductingqubit is increased, then the state of the qubit will transition throughanticrossing 915, and a dip in the voltage across the tank circuit isobserved. Similarly, when the superconducting qubit is in the |1> groundstate, e.g. in region 911, and the flux applied to the superconductingqubit is decreased, then the state of the qubit will transition throughanticrossing 915, and a dip in the tank circuit voltage 810 is observed.Alternatively, when the superconducting qubit is in the |0> groundstate, e.g. in region 910, and the flux applied to the superconductingqubit is decreased, the quantum state of the qubit will not transitionthrough anticrossing 915, and no dip in the tank circuit voltage 810 isobserved. Likewise, if the superconducting qubit is in the |1> groundstate, e.g. in region 911, and the flux applied to the superconductingqubit is increased, the state of the qubit will not transition throughthe anticrossing 915, and no dip in the tank circuit voltage isobserved.

FIG. 9B illustrates an energy level diagram for a qubit in a quantumsystem 850 having an energy level crossing. An energy level crossingarises between energy levels of the qubit when there is no tunnelingterm, a minimal tunneling term, or, more generally, no transition termbetween energy levels of the qubit being read out.

This is in contrast to FIG. 9A, where there is an anticrossing betweenenergy levels of a qubit due to a tunneling term or, more generally, atransition term between such energy levels. Energy levels 950 and 951 ofFIG. 9B lie in part where asymptotes 914 and 916 of FIG. 9A cross. Thecomputational basis states of the qubit represented by the energy leveldiagram FIG. 9B are labeled. The |0> state corresponds to the entireenergy level 950 between crossing 960 and terminus 952. The |1> statecorresponds to the entire energy level 951 between crossing 960 andterminus 953.

At termini 952 and 953, the states corresponding to energy levelsdisappear and the state of the qubit falls to the first available energylevel. As illustrated in FIG. 9B, in the case of the |0> state,beginning at crossing 960, as the flux Φ in the qubit is decreased andthe energy of the |0> state rises, the qubit remains in the |0> stateuntil terminus 952 is reached at which point the |0> state vanishes. Incontrast, if the qubit was in the |1> state, then beginning at crossing960, as the flux Φ in the qubit is decreased, the energy of the |1>state gradually decreases. Thus, the behavior of a qubit that has nocoupling term exhibits hysteretic behavior meaning that the state of thequbit at crossing 960 depends on what the state the qubit was in priorto the flux being brought to the value correlated with crossing 960.

The readout 407 (FIG. 4), in accordance with some embodiments of thepresent invention, of a superconducting qubit having either no couplingterm or a coupling term that is small enough to be disregarded, involvessweeping the flux applied to the superconducting qubit by an amount andin a direction designed to detect crossing 960 and termini 952 and 953.This method makes use of the fact that a tank circuit can detect thecrossing by the hysteretic behavior of the qubit. The measurablequantities are two voltage dips of tank circuit 810 having asymmetricalshape with respect to bias flux. In other words, the tank circuit 810will experience a voltage dip to the left and to the right of crossing960, regardless of which energy state the superconducting qubit is in.Each of the voltage dips is correlated with a state of quantum system850.

Referring to FIG. 9B, if the superconducting qubit is in the |0> state,e.g. at energy level 950, and the flux applied to the superconductingqubit is increased, then the state of the qubit will remain as |0> andno voltage dip is observed. Likewise, if the superconducting qubit is inthe |1> state, e.g. on energy level 951 , and the flux applied to thesuperconducting qubit is decreased, then the state of the qubit willremain as |1> and no voltage dip is observed.

Now consider the case in which the superconducting qubit is in the |0>state, e.g. at energy level 950, and the flux applied to thesuperconducting qubit is decreased. In this case, the qubit will remainin the |0> state until the flux is decreased to just before the pointcorresponding to terminus 952, at which point a wide dip will occur inthe voltage of tank circuit 810 due to slight curvature of level 950 inthe vicinity of terminus 952. After the flux is decreased past terminus952, the state transitions from |0> to |1> because state |0> no longerexists. Consequently, there is an abrupt rise in voltage across tankcircuit 810.

Further consider the case in which a qubit within quantum system 850 ofFIG. 8 is in the |1> state, e.g. at energy level 951 of FIG. 9B, and theflux applied to the superconducting qubit is increased. In this case,the state of the qubit will remain in the |1> basis state until the fluxis increased to just before terminus 953, where a wide dip will occur inthe voltage of tank circuit 810 due to slight curvature in energy level951 in the vicinity of terminus 953. After the flux is increased pastterminus 953, the state transitions to the |0> state and there is anabrupt rise in the voltage of tank circuit 810.

In some embodiments, there is an additional magnetic field B_(A) appliedto the qubit within quantum system 850 of FIG. 8. The sign and magnitudeof this additional magnetic field B_(A) is manipulated so as to read outthe quantum state of qubit 850 by making use of various possible casesof qubit state in relation to location of crossing or anticrossing asdetailed herein. The additional magnetic field B_(A) is applied by thebias lines on the qubit, see, for example, lines 507 of FIG. 5B, or byexcitation device 820 of FIG. 8. The additional magnetic field B_(A)generates the additional flux Φ_(A) according to the formulaΦ_(A)=B_(A)×L_(Q), where L_(Q) is the inductance of the qubit.

The objective of the qubit measurements described in the precedingparagraphs is not to establish the state (ground or excited) that aqubit in quantum system 850 is in, but rather what is the direction ofthe qubit's circulating current (e.g. clockwise 102-0 orcounterclockwise 101-2 as defined in FIG. 1A). The Hamiltonian of anisolated superconducting qubit in its ground state is:H _(Q)=½Δσ_(X)−½εσ_(Z)where Δ and ε are respectively the tunneling and bias energies. Thestates |0> and |1> are assigned to direction of current circulation inthe qubit, for example in the manner illustrated in FIG. 1A or in amanner that is opposite that depicted in FIG. 1A, and are theeigenvectors of the σ_(Z) Pauli matrix. The magnitude of bias energy εdepends on the magnitude of the external magnetic field that is appliedon the qubit. Many adiabatic algorithms have final states where allqubits are strongly biased, so that the tunneling energy term Δ can beneglected. In other words, in many adiabatic algorithms the final statesof the qubits involved in the algorithms have Δ<<ε, meaning that theterm Δ is negligible when compared to the magnitude of ε. When this isthe case, a sweep through appropriate an appropriate range of appliedmagnetic field strengths will yield voltages by tank circuit 810 thatare determined by FIG. 9B rather than FIG. 9A.

The addition of an external magnetic field that is applied against thesuperconducting qubit induces a third additional flux Φ_(A) in thequbit. The total flux applied to the qubit or bias is described as:Φ_(E)=Φ_(DC)+Φ_(RF),where Φ_(DC)=Φ_(A)+f(t)Φ₀.

In some embodiments, of the present invention, the total flux in thequbit is varied in a triangular pattern, with the magnitude and sign ofthe flux chosen such that the states of the qubit can be differentiatedand the distance to anticrossings or crossings can be found. Details onhow such waveforms can accomplish this task are described in Section5.3.2.2, below. In an embodiment of the present invention, the flux isapplied to a superconducting qubit through individual bias wires. Insuch an embodiment each qubit, or all qubits except for one qubit, inquantum system 850 has a bias wire.

5.3.2.2 Waveforms of the Additional Flux

FIG. 10 depicts examples of the waveforms that additional flux Φ_(A)present in a superconducting qubit, or plurality of superconductingqubits, can adopt. This additional flux Φ_(A) can be present in thequbit, for example, due to the application of an external magnetic fieldas described in Section 5.3.2.1. The use of waveforms for additionalflux Φ_(A) permit the user to locate crossings and anticrossings, aswell as to average the readout signal. The location of crossings andanticrossings is proportional to the amplitude of the additional flux.The oscillatory nature of the waveform permits signal average across agiven region of additional flux.

There are independent parameters that can be altered for each waveform.These parameters include, for example, waveform shape, direction,period, amplitude, amplitude growth function, and equilibrium point. Thewaveforms can oscillate in both directions about an equilibrium pointwith a mean of zero (bidirectional), or oscillate away from theequilibrium point in one direction with a non-zero mean(unidirectional). The amplitude for the additional flux Φ_(A) can beconstant, or grow with time. The growth can be continuous or punctuated.The equilibrium point can be a fixed value of flux or vary with time.The shape of the waveform is an oscillatory function that can beselected from a variety of shapes including, for example, sinusoidal,triangular, and trapezoidal, and the low harmonic Fourier approximationsthereof. Exemplary waveforms that can be used in accordance with thepresent invention are illustrated in FIGS. 10A-D. The waveforms, inFIGS. 10A-D are plotted as the additional flux Φ_(A) against time, inarbitrary units.

In an embodiment of the present invention, the maximum amplitude of theadditional flux Φ_(A) in a superconducting qubit is between about 0.01Φ₀ and about 1 Φ₀. In another embodiment of the present invention, themaximum amplitude of the additional flux in a superconducting qubit isbetween about 0.1 Φ₀ and about 0.5 Φ₀. In another embodiment of thepresent invention, the maximum amplitude of the additional flux in asuperconducting qubit varies but has a mean of about 0.25 Φ₀ and candeviate from this value by as much as 0.125 Φ₀. For a superconductingqubit such as qubit 101 of FIG. 1A, having a loop that has a width ofabout 7 μm and a length of about 15 μm, or an inductance of about 30picohenries, this corresponds to a magnetic field of 0.05 Gauss, but candeviate from this value by as much as 0.025 Gauss. In some embodimentsof the present invention, the term “about” in relation to flux valuesdenotes a tolerance that allows for up to a ±50% deviation from thequoted value. In other embodiments, the term “about” in relation to fluxvalues denotes a tolerance that allows for up to ±15%, up to ±5%, up to±2%, or up to ±1% deviation from the quoted value.

The frequency of oscillation, f_(A), is such that the change in fluxΦ_(DC) in quantum system 850 remains adiabatic. In embodiments of thepresent invention, the period ranges from one cycle per second to about100 kilocycles per second. Failure of the process to remain adiabaticresults in a Landau-Zener transition. For more information onLandau-Zener transitions see, for example, Garanin et al., 2002, Phys.Rev. B 66, 174438, which is hereby incorporated by reference in itsentirety.

In some embodiments, rather than sweeping through a range of appliedfluxes in accordance with a single waveform (oscillation, cycle), thewaveform is repeated a plurality of time (plurality of oscillations,plurality of cycles) and voltage response across each of theseoscillations (cycles) is averaged. The number of oscillations needed toperform readout, in such embodiments, depends on the architecture ofquantum system 850. In one embodiment of the present invention, between1 cycle and 100 cycles are used to readout the state of a qubit. In oneembodiment of the present invention, a single cycle that transitionsthrough the Landau-Zener anticrossing (FIG. 9A) or sweeps through ahysteresis loop (FIG. 9B) at a crossing, can perform a readout (step 407of FIG. 4). In other embodiments of the present invention, ten or morecycles (e.g. 20 sweeps through the Landau-Zener anticrossing, FIG. 9A,or 20 sweeps through a hysteresis loop, FIG. 9B) are used to readout thestate of a qubit. In some embodiments of the present invention, up to100 cycles (sweeps) are used to readout the state of a qubit. Morecycles improve the signal to noise ratio of the readout. The number ofcycles needed depends on the total RC characteristic time of theelectronics attached to tank circuit 810 (FIG. 8). When using a lockinamplifier, a preamplifier, an amplifier, a lockin voltmeter, anoscilloscope or any combination thereof, as a tank circuit 810, acombined RC value is achieved that permits readout in a few cycles, oreven a single cycle (e.g., a single sweep through the Landau-Zeneranticrossing, FIG. 9A, or a single sweep through a hysteresis loop, FIG.9B).

FIG. 10A depicts an example of the additional flux Φ_(A) that is appliedto a superconducting qubit, or plurality of superconducting qubits, inaccordance with an embodiment of the present invention. FIG. 10A depictsa unidirectional triangular waveform that grows continuously with time.The value of the flux oscillates with a period that is fixed, but withan amplitude that grows with time. The amplitude can grow linearly,polynomially, logarithmically, exponentially, etc. provided that it doesnot cause the additional flux to grow at such a rate that the change offlux is diabatic. The amplitude is directed to grow when no dip in thevoltage of tank circuit 810 has been observed in a previous cycle sothat more parameter space, e.g. a greater region of addition fluxvalues, can be probed. The amplitude is directed to grow until acrossing (FIG. 9B) or anticrossing (FIG. 9A) is found or the operatordetermines neither is proximate to the parameter space being searched.In FIG. 10A, the additional flux is returned to the equilibrium point ofzero additional flux with every period (every cycle). As shown in FIG.10A, the equilibrium point can be fixed.

FIG. 10B depicts another example of a waveform for additional flux Φ_(A)that is applied to a superconducting qubit, or plurality ofsuperconducting qubits, in accordance with an embodiment of the presentinvention. FIG. 10B depicts a unidirectional triangular waveform theamplitude of which grows in steps. The amplitude can grow by a fixedincrement after a set number of cycles (e.g., 2 or 3). In an embodimentof the present invention, the amplitude of the additional flux grows byup to 1, up to 2, up to 5, up to 10, up to 20, or up to 50 percent witheach step. The amplitude of the additional flux in FIG. 10B grows by upto about 10 percent with every step. The change in amplitude is selectedsuch that the additional flux does not grow at such a rate that thechange of flux is diabatic. The period between steps can be about 2.5cycles (as shown) or every 5, 10, 20, 50, 100, 200, cycles. Thetriangular pattern of FIG. 10B, has peaks that are less sharp than thosein FIG. 10B. In an embodiment of the present invention, the wave formrepresented in FIG. 10B is a low harmonic Fourier approximation of atriangular waveform. As shown in FIG. 10B, the equilibrium point isfixed in some embodiments of the present invention.

FIG. 10C depicts another example of a waveform of additional flux Φ_(A)that is applied to a superconducting qubit, or plurality ofsuperconducting qubits in a quantum system 850, in accordance withanother embodiment of the present invention. FIG. 10C depicts aunidirectional low harmonic approximation of a trapezoidal waveform withan amplitude that grows continuously. In an embodiment of the presentinvention, the number of harmonics, per basis function, in anapproximating waveform is as low as about 1, and as high as about 100.

As shown in FIG. 10C, the equilibrium point can vary with time. Becausethe objective for application of additional flux Φ_(A) is to seek aninterval of flux that locates the anticrossings (FIG. 9B) or hysteresisloop (FIG. 9B), there is no need to re-probe previously probedintervals. Therefore, the equilibrium point of the unidirectionalwaveform can move with time. The amplitude of the additional flux forany given period should be such that the anticrossing (FIG. 9A), orhysteresis loop (FIG. 9B, e.g., the transition from |0> to |1> whencurve 952 is traversed from right to left until state |0> vanishes), canbe resolved. Therefore, the amplitude of the applied flux waveformshould exceed the width of the anticrossing, or hysteresis loop asplotted as a function of flux. Moving the equilibrium point of theunidirectional waveform saves time, while allowing the search to remainadiabatic.

FIG. 10D depicts another example of a waveform of additional flux Φ_(A)that is applied to a superconducting qubit, or plurality ofsuperconducting qubits, in accordance with another embodiment of thepresent invention. FIG. 10D depicts a bidirectional sinusoidal waveformwith amplitude that grows continuously. In FIG. 10D, the equilibriumpoint does not move. Bidirectional waveforms in accordance with FIG. 10Dsearch for an anticrossing (FIG. 9A) or hysteresis loop (FIG. 9B) in twodirections lessening the required accuracy needed to find the locationsof such events.

5.3.2.3 Form of Readout Signal

In accordance with some embodiments of the present invention, FIG. 11depicts examples of the form of readout signals, e.g. tank circuit 810voltage dips, obtained by measurement of a superconducting qubit orplurality of superconducting qubits in a quantum system 850. FIG. 11Aillustrates the form of a readout signal for a superconducting qubitthat includes an anticrossing (FIG. 9A) between two energy levels. FIG.11B illustrates the form of a readout signal for a superconducting qubitthat includes a crossing (FIG. 9B) between two energy levels. FIGS. 11Aand 11B plot the voltage response of a tank circuit against theadditional flux Φ_(A) in the qubit.

FIG. 11A is the output from an oscilloscope. The background noise forthe signal, once averaged over about twenty cycles of a waveform ofadditional flux Φ_(A), is shown as signal 1183. The equilibrium point ofthe additional flux Φ_(A) is denoted line 1182. In the case of abidirectional search, line 1182 shows the average value of theadditional flux applied to the qubit. For a unidirectional search, line1182 shows the minimum amount of additional flux for a period of awaveform from FIG. 10. In many instances, such as the waveform depictedin FIG. 10C, line 1182 and the equilibrium point it represents can moveover time. In an embodiment of the present invention, the input to theoscilloscope is a lock-in voltmeter.

Voltage dips 1180 and 1181 in FIG. 11A are correlated with ananticrossing located to the right and left of equilibrium 1182. Inaccordance with the conventions of FIG. 9A, voltage dips 1180 and 1181corresponds to the measured qubit being in the |0> and |1> state,respectively. Using the conventions of FIG. 9A, the graph illustrated inFIG. 11A depicts two measurement results. If the dip 1180 is observedthis is an indication that the qubit was in the |0> state. If the dip1181 is observed this is an indication that the qubit was in the |1>state. Both dips are drawn for illustrative purposes, but ordinarilyonly one would be observed. A person having ordinary skill in the artwill appreciate that assignment of the labels “0” and “1” to states |0>and |1> is arbitrary and that direction of circulation of the current inthe superconducting qubit is the actual physical quantity beingmeasured. Further a person having ordinary skill in the art can makesuch a labeling for the physical state of any described embodiment ofthe present invention.

FIG. 11B is also the output from an oscilloscope. Graph 11B spans awider range of additional fluxes Φ_(A) than graph 11A. The periodicbehavior of the measured qubit is shown by the repetition of featuresevery flux quantum along the horizontal axis. A voltage dip in this viewis denoted by 1190-1 and corresponds to an unspecified state that is anequilibrium point. Also found in FIG. 11B are voltage dips thatrepresent an energy level crossing for the measured qubit. The plotshows dips 1191-1 and 1192-1, with the tell tale signs of hystereticbehavior—a voltage dip that is wide on one side and has a sharp rise onthe other. As the flux is applied to the qubit, the characteristics ofthe sides reverse location. Hysteric behavior is a term used to describea system whose response depends not only on its present state, but alsoupon its past history. Hysteric behavior is shown by the dips in thevoltage of tank circuit 810 illustrated in FIG. 11B. The behavior atparticular points in the sweep illustrated in FIG. 11B depends onwhether the flux is being increased or decreased. A person havingordinary skill in the art will appreciate that this behavior evidenceshysteretic behavior and therefore the presence of an energy levelcrossing (e.g., crossing 960 of FIG. 9B). See, for example, U.S. patentPublication, US 2003/0224944 A1, entitled “Characterization andmeasurement of superconducting structures,” which is hereby incorporatedby reference in its entirety. Referring to FIG. 11B, the state the qubitwas in prior to readout (prior to applying the flux in accordance with awaveform such as any of those depicted in FIG. 10) is determined by thelocation of voltage dips 1191-1 and 1192-1 relative to an equilibriumpoint. As per the conventions of FIG. 9B, if voltage dips 1191-1 and1192-1 are to the left of the equilibrium point, than the qubit is inthe |1> basis state prior to readout, whereas the qubit is in the |0>basis state if the dips are to the right of the equilibrium point. Aperson having ordinary skill in the art will appreciate that assignmentof the labels “0” and “1” for states |0> and |1> is arbitrary and thatthe direction of circulation of the current in the superconducting qubitis the actual physical quantity being measured.

In an embodiment of the present invention, the voltage dips are in factpeaks because the polarity of the leads to tank circuit 810 wires havebeen reversed. In an embodiment of the present invention, a qubit inquantum system 850 does not have a crossing or an anticrossing in theregion that is probed. The signal from the readout of such a qubit isillustrated as element 1199 in FIG. 11B. In an embodiment of the presentinvention, the state of a qubit can be determined by the relativeposition of the qubit's readout signal to the equilibrium point.

The fidelity of the readout (step 407 of FIG. 4) is limited by theoccurrence of Landau-Zener transitions. A Landau-Zener is a transitionbetween energy states across the anticrossing illustrated in FIG. 9A. Aperson having ordinary skill in the art will appreciate that the mereoccurrence of Landau-Zener transitions does not necessarily conveyinformation about the state of a qubit prior to readout, particularly inadiabatic readout processes. Rather, the exact form and phase of suchtransitions conveys such state information. A Landau-Zener transitionwill appear as a wide, short dip on the oscilloscope screen using theapparatus depicted in FIG. 8. However, the occurrence of a Landau-Zenertransition is only useful if the readout process (the sweep through anapplied flux range) occurring is diabatic. The processes that can bediabatic are certain embodiments of the readout process for a smallnumber of qubits.

In general, and especially during adiabatic evolution for adiabaticquantum computation (transition 404 of FIG. 4), a Landau-Zenertransition should not be permitted to occur in preferred embodiments ofthe present invention. In other words, the occurrence of a Landau-Zenertransition is useful for some configurations of quantum system 850 (FIG.8) in some read out embodiments (step 407) if such read out processesare diabatic. However, in principle, Landau-Zener transitions should notoccur during steps 401, 403, 404, and 405 of FIG. 4 and, in fact, theprobability that such a transition will occur in step 404 serves tolimit the time in which quantum system 850 can be adiabatically evolvedfrom H₀ to H_(P).

One embodiment of the present invention makes use of a negative feedbackloop technique to ensure that Landau-Zener transitions do not occurduring adiabatic evolution 404 (FIG. 4). In this feedback technique, theuser of an adiabatic quantum computer observes the readout from one ormore superconducting qubits undergoing adiabatic evolution. If ananticrossing is approached too fast, tank circuit 810 coupled to quantumsystem 850 (FIG. 8) will exhibit a voltage dip. In response, the user,or an automated system, can repeat the entire process depicted in FIG. 4but evolve at a slower rate during step 404 so that evolution 404remains adiabatic. This procedure permits the adiabatic evolution tooccur at a variable rate, while having a shorter duration, and remain anadiabatic process.

In some embodiments of the present invention, the change in magnitude ofthe response of tank circuit, χ, ranges from 0.01 radians to about 6radians for the phase signal. In some embodiments of the presentinvention, the change in magnitude of the response of the tank circuit,tan(χ), ranges from 0.02 microvolt (μV) to about 1 μV for the amplitudesignal.

5.3.2.4 Adiabatic Readout

Embodiments of the present invention can make use of an adiabaticprocess to readout the state of the superconducting qubit duringmeasurement step 407. Additional flux Φ_(A) and rf flux Φ_(RF) are addedto the superconducting qubit and are modulated in accordance with theadiabatic processes described above. In instances where an additionalflux generates a dip in the voltage of tank circuit 810 and thisadditional flux exceeds the amount of flux needed to reach theequilibrium point of the qubit, the qubit is deemed to have been in the|0> quantum state at the beginning of measurement step 407, inaccordance with the conventions of FIG. 9B. Conversely, in instanceswhere such additional flux generates a dip in the voltage of tankcircuit 810 and this additional flux is less than the amount ofadditional flux needed to reach the equilibrium point of the qubit, thequbit is deemed to have been in the |1> at the beginning of measurementstep 407, in accordance with the conventions of FIG. 9B. The voltage dipis proportional to the second derivative of the energy level withrespect to flux, or other parameters for other qubits. Therefore, thedip occurs at and around the anticrossing 960 where the curvature of theenergy level is greatest. After reading out the state of thesuperconducting qubit, the qubit is returned to its original state. Thatis the state it was in at the beginning of measurement, e.g. the stateunder H_(P). The result of the readout is recorded as a part of step407. The superconducting qubit is returned to its original state, e.g.the state under H_(P), by adiabatically removing the additional flux inthe qubit. In preferred embodiments, the adiabatic nature of this typeof readout does not alter the state of the superconducting qubit. Inother words, the readout does not leave the qubit in a state that isdifferent than the state the qubit was in prior to commencement of thereadout process.

5.3.2.5 Diabatic Readout

In contrast to the example provided in Section 5.3.2.4, embodiments ofthe present invention in accordance with this section make use of adiabatic process to readout the state of a superconducting qubit afterthe adiabatic quantum computation has been completed. In typicalembodiments, this is the only part of the process illustrated in FIG. 4that can be diabatic. In such instances, as part of measurement step407, additional flux Φ_(A) and rf flux Φ_(RF) is added to thesuperconducting qubit with a modulation that is faster than prescribedfor an adiabatic processes using the techniques described above. In suchinstances, when an applied additional flux causes a voltage dip in tankcircuit 810 to occur and such applied additional flux is more than theamount of applied additional flux necessary to achieve the equilibriumpoint of the qubit, the qubit is deemed to have been in the |0> state atthe beginning of measurement step 407, in accordance with theconventions of FIG. 9A. Correspondingly, when an applied additional fluxcauses a voltage dip to occur in tank circuit 810 and this additionalflux is less than the amount of additional flux that is necessary toachieve the equilibrium point for the qubit, the qubit is deemed to havebeen in the |1> state at the beginning of measurement step 407, inaccordance with the conventions of FIG. 9A. The result of the readout isrecorded as a part of step 407. Unlike the embodiments described inSection 5.3.2.4, the diabatic nature of this type of readout can causethe state of the superconducting qubit to alter during readout 407,thereby leaving the qubit in a different state at the end of step 407than the qubit was in at the beginning of step 407.

5.3.2.6 Repeated Readout

Embodiments of the present invention can make use of repeated adiabaticprocesses to readout the states of a plurality of superconducting qubitsin quantum system 850 (FIG. 8). Such embodiments work by reading thestate of each superconducting qubit in succession. In other words, inembodiments where quantum system comprises a plurality of qubits, eachqubit in the plurality of qubits is independently readout in asuccessive manner so then, when any give qubit in the plurality ofqubits in the quantum system 850 is being readout all other qubits inthe quantum system are not being readout. A qubit that is being readoutwhile other qubits in the quantum system are not being readout isreferred to in this section as a target qubit.

In preferred embodiments, each qubit in quantum system 850 (FIG. 8) isread out adiabatically such that the quantum state of each of theremaining qubits in the quantum system is not altered. In contrast tothe single qubit embodiments described above, the state of all thequbits, target and other, are not altered during the readout process 407of any give target qubit. The target qubit's state (the qubit that isbeing readout) is temporarily flipped but the qubit is returned to itsoriginal state, e.g. the state under H_(P), by adiabatically removingthe additional flux in the target qubit. This contributes amultiplicative factor, based on the number of qubits in quantum system850, to the length of the adiabatic computation time. However, anysingle multiplicative factor keeps the overall adiabatic computationtime polynomial with respect to the number of qubits. In someembodiments, quantum system 850 comprises two or more qubits, three ormore qubits, five of more qubits, ten or more qubits, twenty or morequbits, or between two and one hundred qubits.

5.3.2.7 Biasing Qubits During Measurement

As part of measurement step 407, each qubit, when it is the targetqubit, in the plurality of superconducting qubits in quantum system 850(FIG. 8) is biased. The magnetic fields for the bias can be applied bythe bias lines proximate on the qubit. The current used to bias thequbit is dependent on the mutual inductance between the qubit and thebias line. In some embodiments of the present invention, currents usedin biasing the qubit have values of between about 0 miliamperes and 2miliamperes, inclusive. Some embodiments of the present invention makeuse of bias values of up to ±20, up to ±50, up to ±90, up to ±120, up to±150, up to ±180, up to ±210, up to ±240, up to ±300, up to ±550, up to±800, up to ±1100, or up to ±1500 nanoamperes. Here, the term “about”means ±20% of the stated value. In such embodiments, a target qubitwithin quantum system 850 is selected. Additional flux Φ_(A) and rf fluxΦ_(RF) are added to the target qubit, and are modulated in accordancewith the adiabatic readout processes described above. In such instances,when the additional flux that produces a voltage dip in tank circuit 810is more than the amount of flux associated with the equilibrium point ofthe qubit, the qubit is deemed to have been in the |0> state at thebeginning of measurement step 407 in accordance with the conventions ofFIG. 9. Similarly, when the additional flux that produces a voltage dipin tank circuit 810 is less than the amount of flux associated with theequilibrium point of the qubit, the qubit is deemed to have been in the|1> state in accordance with the conventions of FIG. 9. After readingout the state of the target qubit, the target qubit is returned to thestate that the qubit was in prior to measurement. The result of thereadout is recorded in vector {right arrow over (O)} as a part of step407. The process is repeated for a new target qubit in the plurality ofqubits until all the qubits have been readout. In some embodiments, whena new target is selected, a randomly selected bias is applied to the oldtarget qubit. Randomization of the order of target qubits and rerunningthe computation helps avoid errors. The adiabatic nature of this type ofreadout typically does not flip the state of the target qubit nor any ofthe superconducting qubits in the plurality of superconducting qubits.

5.3.2.8 The Quantum Mechanical Nature of the Readout Signal

Readout using tank circuit 810, unlike readout using a DC-SQUID or amagnetic force microscope MFM, is a quantum non-demolition read out(QND). The readout is quantum non-demolition in nature because the qubitremains in its ground state for the entire duration of the readout. Thatis, the qubit remains in the ground state (lower energy level 909 ofFIG. 9A). The qubit may be in some mixture of basis states, such as,(2)_(−1/2)(|0>+|1>) but not (2)^(−1/2)(|0>−|1>) which corresponds to alevel in the exited state (upper energy level 919 of FIG. 9A). Theoutput signal of tank circuit 810 contains information about theamplitude of the states of the superconducting qubit, but collects noinformation about the phase of the oscillating circulating current. Inother words, tank circuit 810 destroys the phase coherence of thepersistent current oscillations of the superconducting qubit during thereadout process leaving the amplitude unperturbed. The reason is thatthe operator probed by tank circuit 810 is σ_(X) and, in this sense,such a readout is the complement of the DC-SQUID and MFM readout, whichmeasures the state of the σ_(Z) operator (Φ_(X) and Φ_(Z) are Paulimatrices). A σ_(X) readout is advantageous in adiabatic quantumcomputing processes. The qubit remains in its ground state also afterthe measurement, e.g., the measurement of one qubit does not perturb theresult of the adiabatic evolution. This allows the readout process to beadiabatic.

5.3.3 Application of Approximate Evaluation Techniques

The detailed exact calculation of energy spectra of an instantaneousHamiltonian H(t) can be intractable due to exponential growth of theproblems size as a function of the number of qubits used in an adiabaticcomputation. Therefore, approximate evaluation techniques are useful asa best guess of the location of the crossings and anticrossings.Accordingly, some embodiments of the present invention make use of anapproximate evaluation method to locate anticrossing of the energies, orenergy spectra, of the qubits in quantum system 850. Doing so lessensthe time needed to probe the anticrossing and crossings with additionalflux.

In an embodiment of the present invention, techniques collectively knownas random matrix theory (RMT) are applied to analyze the quantumadiabatic algorithm during readout. See Bougerol and Lacroix, 1985,Random Products of Matrices with Applications to Schrödinger Operators,Birkhäuser, Basel, Switzerland; and Brody et al., 1981, Rev. Mod. Phys.53, p. 385, each of which is hereby incorporated by reference in itsentirety.

In another embodiment of the present invention, spin density functionaltheory (SDFT) is used as an approximate evaluation method to locateanticrossing of the energies of the system. The probing of the energyspectra by the additional flux can then be used to locate the crossingsand anticrossing and perform a readout of the state of superconductingqubits.

In another embodiment of the present invention, the approximateevaluation technique comprises a classical approximation algorithm inorder to solve NP-Hard problems. When there is a specific instance of aproblem to be solved, the problem is mapped to a description of thequbits being used to solve the NP-Hard problem. This process involvesfinding an approximation algorithm to the problem being solved by thequantum computer and running the approximation algorithm. Theapproximate solution is then mapped to the quantum computer's stateusing the mapping that was used to encode the instance of the NP-Hardproblem. This provides a good estimate for the state of thesuperconducting qubits and lessens the requirements on probing theenergy levels for crossings and anticrossings. Such a mapping typicallyinvolves setting the coupling energies between the qubits being used tosolve the NP-Hard problem so that the qubits approximately represent theproblem to be solved. For examples of approximation algorithms usefulfor the present invention see Goemans and Williamson,“0.878-approximation algorithms for MAX CUT and MAX 2SAT,” InProceedings of the Twenty-Sixth Annual ACM Symposium on the Theory ofComputing, pages 422-431, Montréal, Québec, Canada, 23-25 May 1994;Hochbaum, 1996, Approximation algorithms for NP-hard problems, PWSPublishing Co., Boston; and Cormen et al., 1990, Introduction toAlgorithms, MIT Press, Cambridge, each of which is hereby incorporatedby reference it its entirety.

5.3.4 Solving the MAXCUT Problem Using a Persistent Current QubitQuantum System

FIG. 5B discloses another persistent current qubit topology inaccordance with the present invention. In system 510, the areasencompassed by the respective superconducting loops of qubits 101 areapproximately equal and the heights (y-axis) and widths (x-axis) areequal. In one embodiment in accordance with FIG. 5B, qubits 101 have anarea of 105 micrometers squared (height of 15 micrometers and width of 7micrometers) arranged with two qubits (101-4 and 101-5) pairedlengthwise and a third (101-6) laid transverse and abutting an end ofqubits 101-4 and 101-5. In one specific embodiment of the presentinvention, separations 501 each independently range from about 0.01microns to about 1 micron. The mutual inductance between qubits 101-4and 101-5 is about 10.5 picohenries. The mutual inductance betweenqubits 101-5 and 101-6 is about 5.2 picohenries and the mutualinductance between qubits 101-4 and 101-6 is about 2.5 picohenries.

In another specific embodiment of the present invention, separations 501each independently range from about 0.01 microns to about 1 micron. Themutual inductance between qubits 101-4 and 101-5 is about 18.7picohenries. The mutual inductance between qubits 101-5 and 101-6 isabout 13.3 picohenries and the mutual inductance between qubits 101-4and 101-6 is about 2.5 picohenries.

In an embodiment of the present invention, each qubit 101 in FIG. 5B hasan individual bias line 507. As shown in FIG. 5B, bias lines 507-4,507-5, and 507-6 are used to respectively apply a localized magneticfield to qubits 101-4, 101-5, and 101-6. The bias lines are useful tobring qubits to a degeneracy point where they naturally tunnel. Tominimize the effect one bias line has on another qubit, the incoming andoutgoing wires of a bias line 507 can be braided such that the in andout leads to bias lines 507 overlap in a repeating pattern in order tocancel the field generated by each lead. The bias lines in such anexample would have a loop at the apex of the line 507 proximate to aqubit to be biased by the line. An alternative method of biasing eachqubit is to use a sub-flux quantum generator. See, for example, U.S.patent application Ser. No. 10/445,096, which is hereby incorporated byreference in its entirety. In some embodiments, the current through eachbias line 507 is tunable and independent of the current on other biaslines. However, if a plurality of superconducting qubits 101 each havethe same surface area, one bias line (one circuit) can be used to bias aplurality of qubits 101 provided that the coupling between the bias lineand each of the superconducting qubits is equal.

The adiabatic quantum computing method of the present inventionprogresses using the system described in FIG. 5B in the same manner asdisclosed in Section 5.3.1. Referring to the adiabatic quantum computingprocess of the present invention as described in FIG. 4, in step 401, acomputational problem is set up in system 510 by adjusting distances 501and the height and widths of the respective superconducting loops ofqubits 101. In step 403, the system is initialized to state H₀ by usingbiasing lines 507. In typical embodiments, a characteristic of initialstate H_(o) is that the basis states of qubits 101 cannot couple. Next,in step 405, system 510 is adjusted adiabatically through instantaneousintermediate states in which the basis states of qubits 101 can couple,to the ground state of H_(P), which represents the solution to thecomputational problem. Then, in step 407, the solution to the problem ismeasured by reading out the state of H_(P). As in system 500 of FIG. 5A,there are eight possible solutions {(0,0,0), (0,0,1), (0,1,0), (1,0,0),(0,1,1), (1,1,0), (1,0,1), and (1,1,1)}. The solution adopted by theground state of H_(P) will represent the solution to the problem.

A specific adiabatic quantum computation that can be performed withsystem 510, an optimization based on an instance of MAXCUT, is shown inFIG. 6A.

Step 401 (preparation). Mathematically, to solve MAXCUT as anoptimization, the maximum of the following edge pay off function issought:${P_{E}\left( \left. s \right\rangle \right)} = {\sum\limits_{i = 1}^{V}{\sum\limits_{j > i}^{V}{{s_{i}\left( {1 - s_{j}} \right)}w_{ij}}}}$The vector |s>={s₁, . . . , s_(i), . . . , s_(|V|)}, where s_(i) iseither 0, or 1, is a labeling of all the vertices denoting which side ofthe partition a given vertex is on. The quantity w_(ij) is the weight ofthe edge. In an alternative case of the MAXCUT problem, the maximum ofthe edge pay off function plus the following vertex pay off function issought:${P_{V}\left( \left. s \right\rangle \right)} = {\sum\limits_{i = 1}^{V}{s_{i}w_{i}}}$In some embodiments of the present invention, the MAXCUT problem isencoded into the Hamiltonian H that represents a set of superconductingqubits (e.g., system 510 of FIG. 5B). The qubits represented thevertices and the couplings represent the edges in the MAXCUT for which asolution is sought.

FIG. 6B illustrates a plot of P_(E)+P_(V) for system 600 of FIG. 6A. Theplot represents vertices 601-1, 601-2, and 601-3 as respective pointsX₁, X₂, X₃ along the X-axis. For example, the point 000 in FIG. 6Brepresents the energy of system 600 when vertices 601-1, 601-2, and601-3 are each in the “0” state, the point 001 represents the energy ofsystem 600 when vertices 601-1 and 601-2 are in the “1” state whereasthe vertex 601-3 is in the “1” state, and so forth. There are edgesbetween all the vertices. The vertices have weights w₁, w₂, and w₃,while the edges have weights w₁₂, w₂₃, and w₁₃. Two cuts 611 and 610 insystem 600 are illustrated in FIG. 6A. Cut 610 places vertex 601-2 inone group and vertices 601-1 and 601-3 in the other group. In thenomenclature used in FIG. 6B, this partition can be labeled “010”because vertex 601-2 is “in” while the other two vertices are “out”. Cut611 places 601-3 in one group and the other vertices in another group.This partition can be labeled “001” because vertex 601-3 is “in”.Although FIG. 6 represents only two cuts, quantum systems having morequbits 601 can be employed in order to solve MAXCUT problems involvingmore cuts.

In an embodiment of the present invention, an instance of MAXCUT ismapped onto a quantum system 850 (e.g., system 510) by setting the biason each respective qubit in the quantum system to a value that isproportional to the weight of the corresponding vertex in the graph tobe solved. System 510 differs from system 500 in that each respectivequbit 101 in system 510 can be biased to a different value using, forexample, lines 507. Further, the qubits are arranged with respect toeach other such that the coupling strength between each respective qubitpair is proportional to the edge weight of the corresponding edge in thegraph to be solved. In a particular embodiment of the present invention,an instance of MAXCUT is mapped onto a quantum system 850 (e.g., system510) as follows. The bias on each qubit is set to a value that is halfthe negative of the weight of each vertex. Further, the qubits arearranged such that the coupling strength between each qubit pair is halfthe edge weight of the edge represented by the qubit pair.

Step 403 (Initialization to H₀). The partition of a graph can be foundusing adiabatic quantum computing. Beginning in the state |1000), whichis the ground state of:${H_{0} = {Q{\sum\limits_{i = 1}^{N}\sigma_{i}^{Z}}}},$the adiabatic quantum computation begins. H₀ corresponds to a largemagnetic field being applied to all the qubits.

Step 405 (Transition to H_(P)). By slowly reducing the magnetic fieldwhile still applying bias to individual qubits proportional to theweights assigned to the qubits, the system is changed to one describedby:$H_{P} = {{\sum\limits_{i = 1}^{N}\left\lbrack {{{- \frac{w_{i}}{2}}\sigma_{i}^{Z}} + {\Delta_{i}\sigma_{i}^{X}}} \right\rbrack} + {\sum\limits_{i = 1}^{N}{\sum\limits_{j > i}^{N}{\frac{J_{ij}}{2}{\sigma_{i}^{Z} \otimes {\sigma_{j}^{Z}.}}}}}}$

Step 407 (Measurement). Instep 407, the state of system 510 isdetermined. To appreciate how measurement of the state of H_(P) occursusing system 510, consider FIG. 6B which depicts the various possibleenergy partitions for system 510. In FIG. 6B, the energies of thevarious partitions are based on the following values of the qubitparameters J₁₂=0.246 Kelvin, J₂₃=0.122 Kelvin, J₁₃=0.059 Kelvin,ε₁=0.085 Kelvin, ε₂=0.085 Kelvin, εe₃=0.29 Kelvin, and Δ_(1,2,3)=0.064Kelvin, where 1 Kelvin=20.836 gigahertz. The horizontal axis is the setof eight possible cuts through the graph. The cuts all differ by onenegation of a digit, i.e., “011” follows “001” and not “010”. Here,“010” is the global minimum with energy −12.5 gigahertz while “101” and“001” are local minima. The adiabatic theorem of quantum mechanicsasserts that the system will not be trapped in the local minima.Therefore, when system 510 is in H_(P), it will by in state “010”. Whatfollows is a description of a way to measure system 510 in order todetermine that the system 510 adopts the solution “010” in H_(P) asopposed to some other solution.

Measurement of the state of system 510 can occur by repeating theadiabatic quantum computing steps once (steps 401, 403, 405, and 407)for each respective target qubit 101 in system 510. At step 407, thetarget qubit (the one to be measured) is left in the state determined byH_(P) but all other qubits 101 in the system are biased to a definitestate, e.g. all up. The biasing of all but the target qubit occurs byuse of the bias lines 507. The state of all the qubits can then bereadout and the state of the target qubit can be inferred from the factthat its state either contributes to or against the field created by allthe other qubits. The process is repeated and a new target qubit isselected.

In some embodiments, steps 401, 403, 405 and 407 do not have to berepeated for each qubit in system 510. If the number of tank circuitsused is greater than one, (e.g., at least one biasing element 507 is onan independent circuit) the process can be parallelized with one targetper tank circuit.

5.3.5 Additional Exemplary Persistent Current Qubit Quantum Systems

FIGS. 5C and 5D represent more extensive quantum systems in accordancewith the present invention. As shown in FIGS. 5C and 5D, arrays ofqubits can be formed. In FIGS. 5C and 5D, each box is a qubit. Arraysthat have bias lines attached to each qubit may have a layout likesystem 515. In some embodiments, arrays of qubits do not utilize biaslines for each qubit, and take a form such as that of FIG. 5D. In suchembodiments, bias can be controlled globally, or sub-flux quantumgenerators can be used.

5.4 Charge Qubit Embodiments 5.4.1 Adiabatic Quantum Computing withCharge Qubits

In accordance with an aspect of the present invention, superconductingcharge qubits can be used in adiabatic quantum computation devices(e.g., in quantum systems 850). In an embodiment of the presentinvention, capacitively coupled superconducting charge qubits can beused for adiabatic quantum computing. In an embodiment of the presentinvention, the charge qubits have a fixed or tunable tunneling term. Inan embodiment of the present invention, the couplings between chargequbits have a fixed or tunable sign and/or magnitude of coupling. Someembodiments of the present invention are operated in a dilutionrefrigerator, where the temperature is about 10 milikelvin. Some quantumsystems 850 of the present invention are operated at a temperature below50 milikelvin. The quantum systems 850 of other embodiments of theinvention are operated below 200 milikelvin. Still others are operatedbelow 4.2 Kelvin.

5.4.2 Superconducting Charge Qubits for AQC

A superconducting charge qubit suitable for use in embodiments of thepresent invention is shown in FIG. 12A. The charge qubit 1220 includesan island of superconducting material 1214. In an embodiment of thepresent invention the island encompasses an area of about 32000 squarenanometers (0.032 square micrometers). Island 1214 is connected to areservoir 1210 through a Josephson junction 1211, with Josephson energyE_(J). In an embodiment of the present invention, the thermal energy(k_(B)7) is less than the charge energy of the superconducting chargequbit (E_(C)). In an embodiment of the present invention, the Josephsonenergy (E_(J)) is greater than the superconducting material energy gap.

In some embodiments, qubit 1220 is made of a film of superconductingmaterial deposited on top of an insulating layer over a ground plane. Inan embodiment of the present invention, the film of superconductingmaterial is a layer of aluminum, a thin (about 400 nanometer) layer ofSi₃N₃ is useful insulating layer above a gold ground plane. In such anexample, a suitable insulating material is aluminum oxide, used alone orin combination with other insulating materials. Reservoir 1210 can bemade of similar materials but is larger than island 1214. In otherembodiments of the present invention, the superconducting material isniobium and the insulating material is aluminum oxide.

In an embodiment of the present invention, island 1214 has an effectivecapacitance to ground of about 600 attofarads and is coupled to thereservoir 1210 through a Josephson junction 1211 with the Josephsonenergy of about 20 micro-electronvolts (0.232 kelvin). Reservoir 1210 isa big island with about 0.1 nanofarad or larger capacitance relative tothe ground plane and is otherwise galvanically isolated from theexternal environment. The qubit states are controlled by electrostaticcontrol pulses. The control pulses can be direct current or alternatingcurrent created by generator 1216. The control pulses can be used toinduce a gate charge on superconducting island 1214.

The energy states of the qubit are controlled by electrostatic controlpulses. The control pulses can be direct current or alternating currentThe control pulses can be used to induce a gate charge onsuperconducting island 1214. In some embodiments, the pulses are appliedto the superconducting charge qubit islands, to the SET islands and tothe trap islands in the various embodiments of the invention. The pulsesare applied across the capacitances coupled to the islands. Someembodiments of the present invention require a pulse generator for qubitmanipulation and readout. Suitable generators include an Agilent E8247Cmicrowave generator (Agilent Technologies, Inc., Palo Alto, Calif.) aAnritsu MP1763C (Anritsu Corporation, Kanagawa, Japan, or AnritsuCompany, Morgan Hill, Calif.).

As shown in FIG. 12B, in an embodiment of the present invention, thecharge qubit 1220 depicted in FIG. 12A is modified by replacing theJosephson junction 1211 of qubit 1220 with a flux modulated DC-SQUID1212 in order to form qubit 1230. In particular, island 1214 of qubit1230 is connected to reservoir 1210 through two Josephson junctions inparallel defining a low inductance loop. The Josephson junctions andloop comprise a DC-SQUID 1212. The Josephson energy E_(J) of qubit 1230is controlled by external magnetic field supplied by a flux coil 1227.The Josephson energy E_(J) of DC-SQUID 1212 can be tuned from abouttwice the Josephson energy of the Josephson junctions that make up theDC-SQUID to about zero. In mathematical terms,$E_{J} = {2E_{J}^{0}{{\cos\left( \frac{\pi\quad\Phi_{X}}{\Phi_{0}} \right)}}}$where Φ_(X) is the external flux applied from coil 1227 to DC-SQUID1212, and E_(J) ⁰ is the Josephson energy of one of the Josephsonjunctions in 1212. The DC-SQUID 1212 can also be refereed to as a splitJosephson junction, or a split junction. If one half a flux quantum isapplied to the split junction the effective Josephson energy is zero,when the Josephson energies of the Josephson junctions are equal. If aflux quantum, or no flux, is applied to the split junction, theeffective Josephson energy is 2E_(J) ⁰, or E_(J), _(MAX). The magneticflux in the split Josephson junctions can be applied by a globalmagnetic field.

The Josephson energies of the two Josephson junctions in DC-SQUID 1212are ideally equal but need not be perfectly equal. The coupling throughthe DC-SQUID is proportional to the difference between the Josephsonenergies of the Josephson junctions divided by their sum. With somefabrication practices it is possible to make this ratio one percent orless. Other fabrication processes can make this ratio one tenth of onepercent or less. For adiabatic quantum computing there is no need forabsolute suppression. Because the superconducting charge qubit are awayfrom the degeneracy point in their final states, the error introduced byunequal Josephson junctions will be small enough to be ignored. This isunlike the circuit model of quantum computing where the mismatch of thetwo Josephson junctions that make up the DC-SQUID leads to errors incomputation. The minimum effective Josephson energy of split Josephsonjunction is E_(J), _(MIN).

5.4.3 Read Out Devices for Superconducting Charge Qubits

As illustrated in FIG. 12A, in an embodiment of the present invention,the superconducting charge qubit is coupled to a readout device 1219(e.g., an electrometer). For many types of readout devices 1219, thereadout device is coupled to superconducting charge qubit by acapacitance 1218. In an embodiment of the present invention, thecapacitance 1218 makes up between 1 and 10 percent of the capacitance ofthe superconducting charge qubit island 1214. A good read out device1219 for a superconducting charge qubit is an electrometer sensitiveenough to detect variations of the charge on the island 1214.Electrometers are well known in the art and include single electrontransistors (SETs), radio frequency SETs (RF-SETs), SET and trapcombinations, and superconducting readout circuits analogues ofmeasurements from quantum electrodynamics. In an embodiment of thepresent invention, single shot readout is used. Examples of such readoutdevices are described herein below, some examples of which are shown inFIGS. 12B-D.

5.4.4 QED Read Out

FIG. 12B illustrates a quantum electrodynamics (QED) readout circuitthat can be used as a readout device. Such circuits are known in theprior art. See, for example, Blais, et al., 2004, Phys. Rev. A 69,062320, which is hereby incorporated by reference in its entirety. In anembodiment of the present invention, these QED readout circuits are usedto readout a superconducting qubit at the end of an adiabatic quantumcomputation. Island 1214 of the superconducting charge qubit iscapacitively coupled to a high quality on-chip transmission lineresonator 1225.

At readout, resonator 1225 is exited to its resonance frequency. Theelectric field produced by resonator 1225 has an antinode at the centerof the resonator to which the qubit is strongly coupled by capacitance1228. It has been demonstrated that the coupling between resonator 1225and the qubit can be made so large that a single photon in thetransmission line can resonantly drive Rabi oscillations in thesuperconducting charge qubit at frequencies in excess of 10 megahertz.See, for example, Wallraff et al., 2004, Nature 431, p. 162, which ishereby incorporated by reference in its entirety.

The rate of coherent exchange of a single excitation between thesuperconducting charge qubit and resonator 1225 is much larger than therate at which the superconducting charge qubit decoheres or the ratephotons in resonator 1125 are lost The qubit transitions frequency, thedifference between the ground and first energy level in the qubit,expressed in frequency units, can be detuned from the resonatorfrequency. With this detuning and with a strong coupling betweenresonator 1225 and the qubit, there is a qubit state-dependent frequencyshift in the resonator transition frequency of resonator 1225. Thisfrequency shift can be used to perform a quantum non-demolition (QND)measurement of the qubit state. In this QND measurement, the amplitudeand phase of a probe microwave is measured. This probe microwave is atthe resonance frequency transmitted through resonator 1225 from port1222 to port 1223. In the detuned case, the resonator also enhances thequbit radiative lifetime by providing an impedance transformation thateffectively filters out the noise of the electromagnetic environment atthe qubit transition frequency. See, Devoret, 2004, arXiv.org:cond-mat/0411174, which is hereby incorporated by reference in itsentirety.

5.4.5 Use of a Trap and SET

A single shot readout device is shown in FIG. 12C. In an embodiment ofthe present invention, these single shot readout devices are used toreadout a superconducting qubit at the end of an adiabatic quantumcomputation. The device includes a trap, an electrometer that is aconventional low-frequency single-electron transistor (SET) 1299 with acapacitance of about 1 femtofarads (1 femtofarads=1000 attofarads) and acharge trap 1233, with a capacitance of about I femtofarads or more,placed between island 1214 and SET 1299. The coupling capacitancebetween island 1214 and SET 1299 is about 30 attofarads. Trap 1233 isconnected to island 1214 through resistive tunnel junction 1217. In someembodiments, resistive tunnel junction 1217 has about 100 megaohms ofdirect current resistance. Trap 1233 is coupled to SET 1299 by acapacitance 1234. In typical embodiments, capacitance 1234 is about 100attofarads. The effective coupling between SET island 1239 and qubitisland 1214 is low. In some embodiments it is about 30 attofarads orlower.

The island of trap 1233 is coupled to a voltage source 1231 bycapacitance 1232. SET island 1239 is coupled to voltage source 1237 bycapacitance 1238. SET island 1239 is isolated from the surroundingcircuitry by Josephson junctions 1241 and 1242. SET 1299 can be biasedby optional current source 1244. The signal from the SET is amplified byamplifier 1246. In some embodiments, amplifier 1246 is the same asamplifier 809 of FIG. 8.

The use of trap 1233 enables the user of the system 1250 to separate intime the state manipulation of the qubit's state and readout processes.In addition, qubit island 1214 becomes electrostatically decoupled fromSET 1299. The qubit relaxation rate induced by the SET voltage noise issuppressed by a factor of about 10⁻⁵ for the capacitance values givenabove. However, since the qubit's ground state is read out, therelaxation time of the system is not a limiting factor. In other words,a long time is available for readout and this can significantly improvethe fidelity. Current SETs have a demonstrated charge sensitivity of aslow as a few 10⁻⁵ e/{square root}{square root over (Hz)}, which meansthat the charge variation of 10⁻⁵ e can be detected in a measurementtime of about 1 second. The measurement precision improves as the squareroot of the measurement time. Because the final states of adiabaticquantum computing are classical, the operator of an adiabatic quantumcomputer using superconducting charge qubits has a long time for readout. This aspect alone makes a charge qubit suitable for use in quantumsystems that are used for adiabatic quantum computing in accordance withthe systems and methods of the present invention.

During operation of the qubit, e.g., the steps of adiabatic quantumcomputing, the island of trap 1233 is kept unbiased, prohibiting chargerelaxation that would arise were charges on qubit island 1214 to tunnelonto the trap. To measure the charge state of island 1214, trap 1233 isbiased by a readout pulse applied to capacitance 1232 by source 1231, sothat if the superconducting charge qubit is in the excited state, anextra Cooper-pair charge tunnels into the trap in a sequentialtwo-quasiparticle process. These quasiparticles, or electrons, are theconstituents of the Cooper-pair and create an excess charge on theisland of trap 1233. This excess charge is then detected by SET 1299.System 1299 can be constructed so that the quasiparticle charge on theisland of trap 1233 does tunnel back to island 1214 of the qubit withappreciable probability. The quasiparticle charge on the island of trap1233 influences island 1239 of SET 1299 through capacitor 1234. See, forexample, Devoret and Schoelkopf, 2000, Nature 406, pp. 1039-1046;Astafiev et al., Phys. Rev. Lett. 93, 267007 (2004); and Astafiev etal., 2004, Phys. Rev. B 69, 180507, each of which is hereby incorporatedby reference in its entirety.

Embodiments of the present invention can be operated without trap 1233,capacitor 1234. In such embodiments, island 1239 of SET 1299 is coupleddirectly to the superconducting charge qubit island 1214. In suchembodiments the single shot readout may not be possible but repeatedmeasurements can establish the final state of the superconducting chargequbit. The rate at which measurements can be cycled in such anembodiment is limited by the RC constant of the read out system. Forresistance values through SET island 1239 across junction 1241 and 1242of about 100 kΩ (kiloohms) and signal cable capacitance 1 nanofarad, thecorresponding RC constant limits the measurement cycle to a fewkilohertz. However, this is not a significant limitation because theadiabatic evolution that created the final state for readout is a slowerprocess.

5.4.6 Use of an RF-SET

A radio frequency single electron transistor coupled to asuperconducting charge qubit is shown in FIG. 12D. In an embodiment ofthe present invention, a radio frequency single electron transistor(RF-SET) can be used to readout a superconducting qubit at the end of anadiabatic quantum computation. The readout device, an RF-SET 1297,includes a SET island 1259 coupled to gate and a resonant circuit 1298.In an embodiment SET island 1259 has a capacitance of about 300attofarads. In other embodiments, SET island 1259 has a capacitanceranging from about 100 attofarads to about 500 attofarads. Thecapacitance of the SET island 1259 is the sum of gate capacitance 1258plus the capacitances of the resistive tunnel junctions 1255 and 1256.In some embodiments, the resistance across tunnel junctions 1255 and1256 ranges between about 2 megaohms and about 350 megaohms. In anembodiment of the present invention, capacitance 1218 contributesbetween about 1 and 10 percent of the capacitance of the superconductingcharge qubit island 1214. The coupling capacitance between qubit island1214 and SET island 1259 can be about 5 to 100 attofarads.

SET island 1259 is coupled to voltage source 1257 by capacitance 1258.In an embodiment of the present invention, the capacitance of capacitor1258 is about 20 attofarads. In some embodiments of the presentinvention, the capacitance of capacitor ranges between about 1 attofaradand 100 attofarads. SET island 1259 is isolated from the surroundingcircuit by resistive tunnel junctions 1255 and 1256. RF-SET 1297 can bevoltage biased by a voltage applied on terminal 1264 relative to ground.SET island 1259 is coupled to inductance 1263 and capacitance 1262 ofRF-SET 1297. The value of inductance 1263 and capacitance 1262 is chosento create a resonance frequency of RF-SET 1297. For example, ifinductance 1263 is 600 nanoheneries and capacitance 1262 is about apicofarad, the resonance frequency is about 330 megahertz. Themeasurement cycle time of system 1270 is proportional to the RC constantof RF-SET 1297. For numerical values given above, and an overall RF-SETresistance of 40 kiloohms, the RC time is about a tenth of amicrosecond, or a cycle of seven megahertz. The value of the inductance1263 can be easily and widely varied in fabrication.

In operation, for superconducting charge qubit read out at the end ofthe adiabatic quantum computation using system 1270 of FIG. 12D, thefollowing occurs. The readout of the charge state of the superconductingcharge qubit is accomplished by monitoring the damping of a resonantcircuit to which SET is connected rather than by measuring either thecurrent or the voltage associated with the SET island 1259. A singlefrequency signal at the resonance frequency of the resonant circuit 1298is directed toward SET island 1259. The reflected signal is amplifiedand rectified. The reflected signal varies as a function of the gatevoltage applied to SET island 1259. The gate voltage can be induced byvoltage source 1257 or the charges on the superconducting charge qubitisland 1214. The latter influence makes the RF-SET an electrometeruseful for qubit readout.

The readout scheme specified above for system 1270 has many advantages.It operates at a speed higher than a normal SET. By using alow-impedance and matched impedance high-frequency amplifier, the straycapacitance of the cabling between the SET island 1259 and the amplifier(not shown) becomes unimportant. The amplifier is physically andthermally separated from RF-SET 1298, and can be optimized without theconstraint of very low power dissipation required for operation atmillikelvin temperatures. Finally, because the readout is performed at ahigh frequency, there is no amplifier contribution to the 1/f noise.See, for example, Duty et al., 2004, Phys. Rev. B 69, 140-503;Schoelkopfet al., 1998, Science 280, pp. 1238-1241; and Aassime et al.,2001, Appl. Phys. Lett. 79, pp. 4031-4033, each of which is herebyincorporated by reference in its entirety.

5.4.7 Variation of Superconducting Charge Qubits Parameters

The values of capacitance, inductance, and Josephson energy given inrelation to superconducting charge qubits above, is provided in order todescribe operable embodiments, but these are not the only values thatlead to operable embodiments. The values can be changed with and withoutthe need to make corresponding changes to the cooperative components.There are freely available and commercially available software that canaid the designer of superconducting charge qubits. Such softwareincludes, but is not limited to, FASTCAP a tool designed to calculatecapacitances of a given layout of superconducting devices. FASTCAP isfreely distributed by the Research Laboratory of Electronics of theMassachusetts Institute of Technology, Cambridge, Mass. See also Naborset al., 1992, IEEE Trans. On Microwave Theory and Techniques 40, pp.1496-1507; and Nabors, September 1993, “Fast Three-DimensionalCapacitance Calculation,” Thesis MIT, each of which is herebyincorporated by reference in its entirety. Another useful tool forvarying the capacitances, inductances, and Josephson energies in theexamples above is JSPICE. JSPICE is a simulator for superconductor andsemiconductor circuits with Josephson junction, and is based on thegeneral-purpose circuit simulation program SPICE. SPICE originates fromthe EECS Department of the University of California at Berkeley andprovides links to the download website for this free software. See alsoWhiteley, 1991, IEEE Trans. On Magnetics 27, pp. 2908-2905; and Gaj, etal., 1999, IEEE Trans. Appl. Supercond 9, pp. 18-38, each of which ishereby incorporated by reference in its entirety. In an embodiment ofthe present invention, the thermal energy (k_(B)7) of the charge qubitis less than the charge energy of the superconducting charge qubit(E_(C)). In an embodiment of the present invention, the Josephson energy(E_(J)) of the charge qubit is greater than the superconducting materialenergy gap of the charge qubit In an embodiment of the presentinvention, the charge energy of the charge qubit is about equal to theJosephson energy of the charge qubit

5.4.8 Coupled Superconducting Charge Qubits

Some embodiments of the present invention comprise a lattice ofinterconnected superconducting charge qubits that are capacitivelycoupled to each other. This is a difference between these charge qubitsand the systems of FIG. 5, which are being inductively coupled. Shown inFIG. 13A is an example of a fixed coupling between two superconductingcharge qubits. Shown in FIG. 13B is an example of a tunable couplingbetween two superconducting charge qubits. The numbers of couplings andqubits in the lattice is scalable.

As shown in FIG. 13A, the couplings between charge qubits have a tunablesign and/or tunable magnitude of coupling. The superconducting chargequbits each have an island 1214-1 and 1214-2, coupled to a reservoir1210, through split Josephson junctions 1212-1, and 1212-2. Disposedbetween the superconducting charge qubit islands, is a couplingcapacitance 1318. The value of the capacitance 1318 is denoted C_(C). Inan embodiment of the present invention, coupling capacitance 1318 has avalue of about 1 attofarad. In other embodiments of the presentinvention, the coupling capacitance is between about 1/100 attofaradsand 50 attofarads. See, for example, Nakamura et al., 2003, Nature 421,pp. 823-826, which is hereby incorporated by reference in its entirety,which describes some charge qubits generally, including operatingparameters of such qubits.

In the embodiment illustrated in FIG. 13B, the coupling 1325 betweencharge qubits have a tunable sign and/or tunable magnitude of coupling.The superconducting charge qubits each have an island coupled to areservoir 1210 through Josephson junctions 1211-3, and 1211-4. Disposedbetween the superconducting charge qubit islands, is a variableelectrostatic transformer (VET), 1325. The VET comprises an island,separated from a voltage source by a Josephson junction. The island iscoupled to the superconducting charge qubit islands it couples with bytwo respective capacitors. The VET is described in Averin and Bruder,2003, Phys. Rev. Lett. 91, 057003, which is hereby incorporated byreference in its entirety.

FIG. 14 illustrates graphs that correspond to the way superconductingcharge qubits can be arranged in quantum systems in accordance withvarious embodiments of the present invention. The qubits used in thesystems of the present invention can be arranged in a nearest neighbourtriangular, rectangular, lattice, etc. with each qubit having 3, 4,etc., neighbors. Such lattices are graphs of degree three, four, etc.The number of neighbors depends on whether the qubit is in the interior,or exterior of the graph. Qubits on the exterior have fewer neighbors.The superconducting charge qubit can be arranged in graphs of higherdegrees. A non-planar graph is a graph that cannot be drawn on a twodimensional plan without two edges crossing.

FIGS. 14A and 14B illustrate planar graphs that correspond to waysqubits can be arranged, in accordance with the present invention. Planargraphs are graphs in which edges do not cross. Graphs 515 and 525 areequivalent to layouts shown in FIG. 5. The graphs are planar, andrectangular. Planar rectangular graphs are well suited for inductivelycoupled qubits. Because the coupling is dictated by geometry andproximity, inductively coupled qubits do not lend well to being arrangedas non-planar graphs.

FIGS. 14D and 14D illustrate non-planar graphs that corresponds to waysqubits can be arranged, in accordance with various embodiments of thepresent invention. Graph 1370 is an example of a graph upon which MAXCUTcan be solved. Other NPC problems can be solved on graphs like 1370 and1390. Capacitance coupling lends it self to creating arrangements thatare equivalent to non-planar graphs. The couplings are mediated bycapacitance, but the wires leading to the capacitor plates can cross.

5.4.9 Mathematical Description of Coupled Superconducting Charge Qubits

The Hamiltonian for lattices of superconducting charge qubits 1220,1230, 1310, and 1330, etc. is:$H = {{{- \frac{1}{2}}{\sum\limits_{i = 1}^{N}\left\lbrack {{E_{Ci}^{\prime}\sigma_{i}^{Z}} + {E_{Ji}^{\prime}\sigma_{i}^{X}}} \right\rbrack}} + {\sum\limits_{i = 1}^{N}{\sum\limits_{j > i}^{N}{J_{ij}{\sigma_{i}^{Z} \otimes \sigma_{j}^{Z}}}}}}$which is similar in form to Hamiltonians given above for coupled phasequbits. All terms in the Hamiltonian are potentially in the initial,final, or instant Hamiltonian of a computational device that isperforming an adiabatic quantum computation in accordance with FIG. 4.Every term in this Hamiltonian can be fixed or tunable.

The biasing term proportional to the σ^(Z) Pauli matrix is tunable. Thebias is applied to the superconducting charge qubit via capacitorcoupled to the qubit. Mathematically the term is:E′ _(C)=4E _(C)(1−2n _(g))where the dimensionless gate charge n_(g)≈C_(g)V_(g)/(2e) is determinedin operation by the gate voltage V_(g) and in fabrication by the gatecapacitance C_(g). The gate capacitance is capacitance 1215. Here it isassumed that n_(g)≈½, but in parts of the computation the dimensionalcharge and hence the state of the qubit is biased to either 1 or 0.

The tunneling term proportional to the σ^(X) Pauli matrix is alwayspresent except for embodiments where the Josephson energy term E_(J) istunable. In an embodiment of the present invention, the charge qubitswith a tunable tunneling term have a Josephson junction connected to theisland of the charge qubit replaced by split Josephson junction. Inthese embodiments, tunneling is suppressed by tuning the flux in thesplit Josephson junction. The suppression of tunneling is absolute whenthe Josephson energies of junction are equal and the flux in theDC-SQUID loop is a half flux quantum. The tunneling is maximal (at zerobias) when the flux in the split Josephson junction is a whole numbermultiple of a flux quantum (Φ₀), including zero. Traditionally, thetunneling of the qubit states is suppressed when the qubits are in thefinal state of an adiabatic quantum computation. Flux through splitJosephson junctions can be tuned by a global magnetic field appliedperpendicular to the plane of superconducting charge qubits. The globalfield can have local corrections applied through flux coils like 1227 ofFIG. 12B.

In embodiments of the present invention, the coupling term proportionalto the tensor product of σ^(Z) Pauli matrix is tunable. In otherembodiments, the coupling term is fixed. Coupling between twosuperconducting charge qubits can have many forms. If the Hamiltonianterm for the coupling has a principal component proportional toσ^(Z){circle over (×)}σ^(Z), then the coupling is suitable for adiabaticquantum computing. An example of a coupling between superconductingcharge qubits that has a principal component proportional toσ^(Z){circle over (×)}σ^(Z), is capacitive coupling. The sign of thiscoupling is positive making it an antiferromagnetic coupling. Thecoupling sign and coefficient (J_(ij)) are not tunable for a simplecapacitance coupling unless a VET is used. The coupling in terms ofphysical parameters is${J_{ij} = \frac{q_{i}q_{j}}{{\overset{\sim}{C}}_{C}}},$where q_(i) and q_(j) is the excess charge on the i^(th) and j^(th)qubit, and${\overset{\sim}{C}}_{C} \equiv \frac{C_{i}^{\Sigma}C_{j}^{\Sigma}}{C_{C}}$where C_(k) ^(Σ) is the total capacitance for the k^(th) qubit. Thecoupling sign and coefficient (J_(ij)) are tunable if a variableelectrostatic transformer (VET) is used. The VET can be used to createferromagnetic and antiferromagnetic interactions between superconductingqubits. The VET is described in Averin and Bruder, 2003, Phys. Rev.Lett. 91, 057003, which is hereby incorporated by reference in itsentirety.

Operation

In accordance with the general procedure of adiabatic quantum computingas shown in FIG. 4, some embodiments of the present invention usesuperconducting charge qubits. In step 401 a quantum system that will beused to solve a computation is selected and/or constructed and includesa plurality superconducting charge qubits. Once the pluralitysuperconducting charge qubits has been configured, an initial state anda final state of the system are defined. The initial state ischaracterized by the initial Hamiltonian H₀ and the final state ischaracterized by the final Hamiltonian H_(P) that encodes thecomputational problem to be solved. The initial Hamiltonian H₀ and thefinal Hamiltonian H_(P) are variants of the Hamiltonian of coupledsuperconducting charge qubits given above making use of the tunableHamiltonian elements. More details on how the Hamiltonian elements aretuned is described hereinabove.

In step 403, the plurality superconducting charge qubits is initializedto the ground state of the time-independent Hamiltonian, H₀, whichinitially describes the states of the qubits. It is assumed that theground state of H₀ is a state to which the plurality superconductingcharge qubits can be reliably and reproducibly set. This assumption isreasonable for plurality superconducting charge qubits including qubitbias terms.

In transition 404 between steps 403 and 405, the pluralitysuperconducting charge qubits are acted upon in an adiabatic manner inorder to alter the states of the plurality superconducting chargequbits. The plurality superconducting charge qubits change from beingdescribed by Hamiltonian H₀ to a description under H_(P). This change isadiabatic, as defined above, and occurs in a time T. In other words, theoperator of an adiabatic quantum computer causes the pluralitysuperconducting charge qubits, and Hamiltonian H describing theplurality superconducting charge qubits, to change from H₀ to a finalform H_(P) in time T. The change is an interpolation between H₀ andH_(P).

In accordance with the adiabatic theorem of quantum mechanics, theplurality of superconducting charge qubits will remain in the groundstate of H at every instance the qubits are changed and after the changeis complete, provided the change is adiabatic. In some embodiments ofthe present invention, the plurality of superconducting charge qubitsstart in an initial state H₀ that does not permit quantum tunneling, areperturbed in an adiabatic manner to an intermediate state that permitsquantum tunneling, and then are perturbed in an adiabatic manner to thefinal state described above.

In step 405, the plurality of superconducting charge qubits has beenaltered to a state that is described by the final Hamiltonian. The finalHamiltonian H_(P) can encode the constraints of a computational problemsuch that the state of the plurality superconducting charge qubits underH_(P) corresponds to a solution to this problem. Hence, the finalHamiltonian is also called the problem Hamiltonian H_(P). If theplurality superconducting charge qubits is not in the ground state ofH_(P), the state is an approximate solution to the computationalproblem. Approximate solutions for many computational problems areuseful and such embodiments are fully within the scope of the presentinvention.

An aspect of the problem Hamiltonian H_(P) is the energy of thetunneling terms in the Hamiltonian, which are either weak or zero. Theenergy of the tunneling terms is suppressed by applying a half fluxquantum through the split Josephson junctions that separate thesuperconducting charge qubit island from the reservoir.

In step 407, the plurality superconducting charge qubits described bythe final Hamiltonian H_(P) is read out. The read out can be in theσ^(Z) basis of the qubits. If the read out basis commutes with the termsof the problem Hamiltonian H_(P), then performing a read out operationdoes not disturb the ground state of the system. The read out method cantake many forms. The object of the read out step is to determine exactlyor approximately the ground state of the system. The states of allqubits are represented by the vector {right arrow over (O)}, which givesa concise image of the ground state or approximate ground state of thesystem. The plurality superconducting charge qubits can be readout usingthe devices and techniques described hereinabove.

Exemplary Embodiment of a Method of AQC with Superconducting ChargeQubits

This example describes an embodiment of the invention where thesuperconducting charge qubits have fixed couplings. To perform adiabaticquantum computation with superconducting charge qubits, in step 401, theoperator chooses E^(′) _(Ci) and J_(ij) in such a way as to simulate theproblem Hamiltonian H_(P). The charging energy E′_(Ci) can be controlledby the gate voltages, and qubit-qubit coupling energy J_(ij) is fixed infabrication by the capacitance of the coupling capacitances unless avariable electrostatic transformer is used. In an embodiment of thepresent invention, these values are chosen to satisfy the constraintE_(J,MIN)<<E′_(Ci) and E_(J,MIN)<<J_(ij) while E′_(Ci)≦E_(J,MAX) andJ_(ij)≦E_(J,MAX), for every qubit or pair of coupled qubits. At thebeginning of the computation, step 403, E_(Ji) are set to E_(J,MAX). Inthat case, the tunneling terms dominate causing a decay of the system tothe ground state. Then, in transition 404, the E_(Ji) terms areadiabatically reduced to E_(J,MIN). At the end of the adiabaticevolution, the tunneling terms are negligible and the effectiveHamiltonian is H_(P). Then, in step 405, the qubits can be read out.Suitable techniques for single shot readout include the use of a trapisland, an RF-SET, QED read out circuits, etc.

Second Example of a Method of AQC with Superconducting Charge Qubits

This example describes an embodiment of the invention where thesuperconducting charge qubits have fixed couplings. In an embodiment ofthe present invention where E′_(Ci) is greater than E_(J,MAX), but notmuch greater, the operator can artificially reduce E′_(Ci). To performadiabatic quantum computation with superconducting charge qubits, step401, the operator chooses J_(ij) in such a way as to simulate theproblem Hamiltonian Hp. In an embodiment of the present invention, thesevalues are chosen to satisfy E_(J,MIN)<<E′_(Ci) and E_(J,MIN)<<J_(ij)while E′_(Ci)≧E_(J,MAX) and J_(ij)≦E_(J,MAX,) for every qubit or pair ofcoupled qubits. At the beginning of the computation, step 403, eachE_(Ji) is set to E_(J,MAX). The gate voltages on the superconductingcharge qubits are tuned to bring the qubits closer to degeneracy[dimensionless gate charge equal to one half (n_(g)=½,)]. This reducesthe energy in the bias term E′_(Ci). Then, in transition 404, the E_(Ji)terms are adiabatically reduced to E_(J,MIN). Also, in the adiabaticevolution the bias on the superconducting charge qubit is adiabaticallyincreased at the same time as each E_(Ji) is decreased. The bias on thesuperconducting charge qubits is changed such that, at the end of theadiabatic evolution, the bias terms are appropriate for the problemHamiltonian is H_(P). Then, in step 405, the qubits are read out.

5.4.10 Further Embodiments for AQC with Superconducting Charge Qubits

Further embodiments for adiabatic quantum computing with superconductingcharge qubits can be constructed from the following protocols for thebiasing, tunneling, and coupling of qubits.

For the tunneling terms of a plurality of superconducting charge qubitsthe following protocols can be used. In an embodiment where thetunneling term is fixed, the bias and coupling energies for the initialHamiltonian are set to be less that the tunneling energies. In the sameexamples, the bias and coupling energies for the problem Hamiltonian areset to be greater that the tunneling energies. In such a case, thetunneling is weak compared with the final Hamiltonian. In embodimentswhere the tunneling term is tunable, the tunneling energies for theinitial Hamiltonian are set to values approaching and includingE_(J,MAX) but away from zero. After the adiabatic evolution thetunneling energies are set to E_(J,MIN), which can be zero.Alternatively, the tunneling energies for the initial Hamiltonian areset to E_(J,MIN), which can be zero. During the adiabatic evolution thetunneling energies are to values approaching and including E_(J,MAX) butaway from zero. After the adiabatic evolution the tunneling energies areset to E_(J,MIN), which can be zero.

For the coupling terms of a plurality of superconducting charge qubitsthe following protocols can be used. In an embodiment where the couplingterms are fixed at fabrication time, the couplings are set such thatthey describe the couplings for the problem Hamiltonian. In anembodiment where the coupling terms are tunable, the coupling energiesfor the initial Hamiltonian are set to zero. During the adiabaticevolution the coupling energies are increased such that they describethe couplings for the problem Hamiltonian. If the tunneling energies areat zero during readout, the coupling energies can be set to any valuewithout the final state of the superconducting charge qubits charging.

5.5 Representative System

FIG. 15 illustrates a system 1500 that is operated in accordance withone embodiment of the present invention. System 1500 includes at leastone digital (binary, conventional) interface computer 1501. Computer1501 includes standard computer components including at least onecentral processing unit 1510, memory 1520, non-volatile memory, such asdisk storage 1515. Both memory and storage are for storing programmodules and data structures. Computer 1501 further include input/outputdevice 1511, controller 1516 and one or more busses 1517 thatinterconnect the aforementioned components. User input/output device1511 includes one or more user input/output components such as a display1512, mouse 1513, and/or keyboard 1514.

System 1500 further includes an adiabatic quantum computing device 1540that includes those adiabatic quantum computing devices shown above.Exemplary examples of adiabatic quantum computing devices 1540 include,but are not limited to, devices 101 of FIG. 1; 500, 510, 515, 520, 525,535, and 555 of FIG. 5; 800 of FIG. 8; 1220, 1230, 1250, and 1270 ofFIG. 12; 1310 and 1330 of FIG. 13; and 515, 520, 1470, and 1490 of FIG.14. This list of exemplary examples of adiabatic quantum computingdevices is non-limiting. A person of ordinary skill in the art willrecognize other devices suitable for 1540.

System 1500 further includes a readout control system 1560. In someembodiments, readout control system 1560 comprises a plurality ofmagnetometers, or electrometers, where each magnetometer or electrometeris inductively coupled, or capacitively coupled, respectively, to adifferent qubit in quantum computing device 1540. In such embodiments,controller 1516 receives a signal, by way of readout control system1560, from each magnetometer or electrometer in readout device 1560.System 1500 optionally comprises a qubit control system 1565 for thequbits in quantum computing device 1540. In some embodiments, qubitcontrol system 1565 comprises a magnetic field source or electric fieldsource that is inductively coupled or capacitively coupled,respectively, to a qubit in quantum computing device 1540. System 1500optionally comprises a coupling device control system 1570 to controlthe couplings between qubits in adiabatic quantum computing device 1540.A coupling that is controllable includes, but is not limited to,coupling 1325 of FIG. 13.

In some embodiments, memory 1520 includes a number of modules and datastructures. It will be appreciated that at any one time during operationof the system, all or a portion of the modules and/or data structuresstored in memory 1520 are resident in random access memory (RAM) and allor a portion of the modules and/or data structures are stored innon-volatile storage 1515. Furthermore, although memory 1520, includingnon-volatile memory 1515, is shown as housed within computer 1501, thepresent invention is not so limited. Memory 1520 is any memory housedwithin computer 1501 or that is housed within one or more externaldigital computers (not shown) that are addressable by digital computer1501 across a network (e.g., a wide area network such as the internet).

In some embodiments, memory 1520 includes an operating system 1521.Operating system 1521 includes procedures for handling various systemservices, such as file services, and for performing hardware dependenttasks. In some embodiments of the present invention, the programs anddata stored in system memory 1520 further include an adiabatic quantumcomputing device interface module 1523 for defining and executing aproblem to be solved on an adiabatic quantum computing device. In someembodiments, memory 1520 includes a driver module 1527. Driver module1527 includes procedures for interfacing with and handling the variousperipheral units to computer 1501, such as controller 1516 and controlsystems 1560, qubit control system 1565, coupling device control system1570, and adiabatic quantum computing device 1540. In some embodimentsof the present invention, the programs and data stored in system memory1520 further include a readout module 1530 for interpreting the datafrom controller 1516 and readout control system 1560.

The functionality of controller 1516 can be divided into two classes offunctionality: data acquisition and control. In some embodiments, twodifferent types of chips are used to handle each of these discretefunctional classes. Data acquisition can be used to measure physicalproperties of the qubits in adiabatic quantum computing device 1540after adiabatic evolution has been completed. Such data can be measuredusing any number of customized or commercially available dataacquisition microcontrollers including, but not limited to, dataacquisition cards manufactured by Elan Digital Systems (Fareham, UK)including, but are not limited to, the AD132, AD136, MF232, MF236,AD142, AD218, and CF241. In some embodiments, data acquisition andcontrol is handled by a single type of microprocessor, such as the ElanD403C or D480C. In some embodiments, there are multiple interface cards1516 in order to provide sufficient control over the qubits in acomputation 1540 and in order to measure the results of an adiabaticquantum computation on 1540.

CONCLUSION AND REFERENCES CITED

When introducing elements of the present invention or the embodiment(s)thereof, the articles “a,” “an,” “the,” and “said” are intended to meanthat there are one or more of the elements. The terms “comprising,”“including,” and “having” are intended to be inclusive and to mean thatthere may be additional elements other than the listed elements.Moreover, the term “about” has been used to describe specificparameters. In many instances, specific ranges for the term “about” havebeen provided. However, when no specific range has been provided for aparticular usage of the term “about” herein, than either of twodefinitions can be used. In the first definition, the term “about” isthe typical range of values about the stated value that one of skill inthe art would expect for the physical parameter represented by thestated value. For example, a typical range of values about a specifiedvalue can be defined as the typical error that would be expected inmeasuring or observing the physical parameter that the specified valuerepresents. In the second definition of about, the term “about” meansthe stated value ±0.10 of the stated value. As used herein, the term“instance” means the execution of a step. For example, in a multistepmethod, a particular step may be repeated. Each repetition of this stepis referred to herein as an “instance” of the step.

All references cited herein are incorporated herein by reference intheir entirety and for all purposes to the same extent as if eachindividual publication or patent or patent application was specificallyand individually indicated to be incorporated by reference in itsentirety for all purposes.

Alternative Embodiments

The present invention can be implemented as a computer program productthat comprises a computer program mechanism embedded in a computerreadable storage medium. For instance, the computer program productcould contain program modules, such as those illustrated in FIG. 15,that implement the various methods described herein. These programmodules can be stored on a CD-ROM, DVD, magnetic disk storage product,or any other computer readable data or program storage product. Thesoftware modules in the computer program product can also be distributedelectronically, via the Internet or otherwise, by transmission of acomputer data signal (in which the software modules are embedded) on acarrier wave.

Many modifications and variations of this invention can be made withoutdeparting from its spirit and scope, as will be apparent to thoseskilled in the art. The specific embodiments described herein areoffered by way of example only, and the invention is to be limited onlyby the terms of the appended claims, along with the full scope ofequivalents to which such claims are entitled.

1. A computer program product for use in conjunction with a computersystem, the computer program product comprising a computer readablestorage medium and a computer program mechanism embedded therein, thecomputer program mechanism comprising: instructions for initializing aquantum system comprising a plurality of superconducting qubits to aninitialization Hamiltonian H_(O), wherein the quantum system is capableof being in one of at least two configurations at any give time, the atleast two configurations including: a first configuration characterizedby the initialization Hamiltonian H_(O), and a second configurationcharacterized by a problem Hamiltonian H_(P), and wherein eachrespective first superconducting qubit in said plurality ofsuperconducting qubits is arranged with respect to a respective secondsuperconducting qubit in the plurality of superconducting qubits suchthat the first respective superconducting qubit and the secondrespective superconducting qubit define a predetermined couplingstrength and wherein the predetermined coupling strengths between eachsaid first respective superconducting qubit and second respectivesuperconducting qubit collectively define a computational problem to besolved; instructions for adiabatically changing the quantum system untilit is described by the ground state of the problem Hamiltonian H_(P);and instructions for reading out the state of the quantum system.
 2. Thecomputer program product of claim 1, the computer program mechanismfurther comprising instructions for repeating said instructions forbiasing, instructions for adding, and instructions for adiabaticallyvarying between 1 time and 100 times inclusive, and wherein the presenceor absence of the voltage response of the tank circuit is observed as anaverage of the voltage response of the tank circuit to each instance ofthe instructions for adiabatically changing that are executed by saidinstructions for repeating.
 3. A computer program product for use inconjunction with a computer system, the computer program productcomprising a computer readable storage medium and a computer programmechanism embedded therein, the computer program mechanism fordetermining a quantum state of a first target superconducting qubit in aplurality of superconducting qubits,.the computer program mechanismcomprising: instructions for initializing a plurality of superconductingqubits so that they are described by a problem Hamiltonian, wherein theproblem Hamiltonian describes (i) the quantum state of the plurality ofsuperconducting qubits and (ii) each coupling energy between qubits inthe plurality of qubits, and wherein the problem Hamiltonian is at ornear a ground state; instructions for adding an rf-flux to the firsttarget superconducting qubit, wherein the rf-flux has an amplitude thatis less than one flux quantum; and instructions for adiabaticallyvarying an amount of an additional flux in the first targetsuperconducting qubit and observing a presence or an absence of a dip ina voltage response of a tank circuit that is inductively coupled withthe first target superconducting qubit during a time when saidinstructions for adiabatically varying are executing
 4. The computerprogram product of claim 3, wherein each superconducting qubit in theplurality of superconducting qubits is in a quantum ground state duringall or a portion of said instructions for initializing, instructions foradding, and said instructions for adiabatically varying.
 5. The computerprogram product of claim 3, wherein the problem Hamiltonian correspondsto a terminus of an adiabatic evolution of the plurality ofsuperconducting qubits.
 6. The computer program product of claim 3,further comprising instructions for biasing all or a portion of thesuperconducting qubits in the plurality of superconducting qubits,wherein the problem Hamiltonian additionally describes the biasing onthe qubits in the plurality of superconducting qubits.
 7. The computerprogram product of claim 6, wherein an energy of said biasing exceedsthe tunneling energy of a tunneling element of the Hamiltonian of asuperconducting qubit in said plurality of superconducting qubitsthereby causing tunneling to be suppressed in the superconducting qubitduring an instance of said instructions for biasing, instructions foradding and said instructions for adiabatically varying.
 8. The computerprogram product of claim 3, the computer program mechanism furthercomprising instructions for adiabatically removing additional flux thatwas added to the first target superconducting qubit during saidinstructions for adiabatically varying.
 9. The computer program productof claim 3, wherein the instructions for adiabatically varying compriseinstructions for adiabatically varying the additional flux in accordancewith a waveform selected from the group consisting of periodic,sinusoidal, triangular, and trapezoidal.
 10. The computer programproduct of claim 3, wherein the instructions for adiabatically varyingcomprise instructions for adiabatically varying the additional flux inaccordance with a low harmonic Fourier approximation of a waveformselected from the group consisting of periodic, sinusoidal, triangular,and trapezoidal.
 11. The computer program product of claim 3, whereinthe additional flux has a direction that is deemed positive or negative.12. The computer program product of claim 3, wherein the instructionsfor adiabatically varying are characterized by a waveform that has anamplitude that grows with time, wherein the amplitude of the waveformcorresponds to an amount of additional flux that is added to the firsttarget superconducting qubit during an instance of said instructions foradiabatically varying.
 13. The computer program product of claim 3,wherein the additional flux has an equilibrium point that varies withtime.
 14. The computer program product of claim 3, wherein theadditional flux is either unidirectional or bidirectional.
 15. Thecomputer program product of claim 3, wherein the additional flux has afrequency of oscillation between about 1 cycle per second and about 100kilocycles per second.
 16. The computer program product of claim 3,wherein the instructions for adding comprise instructions for addingsaid rf-flux using (i) an excitation device that is inductively coupledto the first target superconducting qubit or (ii) the tank circuit 17.The computer program product of claim 3, the computer program mechanismfurther comprising instructions for repeating said instructions foradding and said instructions for adiabatically varying between 1 timeand 100 times, and wherein the presence or absence of the voltageresponse of the tank circuit is observed as an average of the voltageresponse of the tank circuit across each instance of the instructionsfor adiabatically varying that is executed by said instructions forrepeating.
 18. A computer system for determining a quantum state of afirst target superconducting qubit in a plurality of superconductingqubits, the computer system comprising: a central processing unit; amemory, coupled to the central processing unit, the memory storinginstructions for biasing all or a portion of the qubits in the pluralityof superconducting qubits other than the first target superconductingqubit, wherein a problem Hamiltonian describes (i) the biasing on thequbits in the plurality of superconducting qubits and (ii) each couplingenergy between respective superconducting qubit pairs in the pluralityof superconducting qubits, and wherein the problem Hamiltonian is at ornear a ground state; instructions for adding an rf-flux to the firsttarget superconducting qubit, wherein the rf-flux has an amplitude thatis less than one flux quantum; and instructions for adiabaticallyvarying an amount of an additional flux in the first targetsuperconducting qubit and observing a presence or an absence of a dip ina voltage response of a tank circuit that is inductively coupled withthe first target superconducting qubit during a time when saidinstructions for adiabatically varying are executed.
 19. A structure foradiabatic quantum computing comprising: a plurality of superconductingqubits, wherein said plurality of superconducting qubits are capable ofbeing in any one of at least two configurations at any give time, the atleast two configurations including: a first configuration characterizedby an initialization Hamiltonian H₀, and a second Hamiltoniancharacterized by a problem Hamiltonian H_(P), the problem Hamiltonianhaving a ground state, wherein each respective first superconductingqubit in said plurality of superconducting qubits is coupled with arespective second superconducting qubit in the plurality ofsuperconducting qubits such that the first respective superconductingqubit and the corresponding second respective superconducting qubitdefine a predetermined coupling strength and wherein the predeterminedcoupling strength between each said first respective superconductingqubit and corresponding second respective superconducting qubitcollectively defines a computational problem to be solved; and a tankcircuit inductively coupled to all or a portion of said plurality ofsuperconducting qubits.
 20. The structure of claim 19 wherein asuperconducting qubit in said plurality of superconducting qubits is apersistent current qubit.
 21. The structure of claim 19 wherein the tankcircuit has a quality factor that is greater than
 1000. 22. Thestructure of claim 19 wherein the tank circuit comprises an inductiveelement and wherein the inductive element comprises a pancake coil ofsuperconducting material.
 23. The structure of claim 22 wherein thepancake coil is made of a superconducting material and comprises a firstturn and a second turn, and wherein the superconducting material of thepancake coil is niobium, and there is a spacing of 1 about micrometerbetween the first turn and the second turn of the pancake coil.
 24. Thestructure of claim 19 wherein the tank circuit comprises an inductiveelement and a capacitive element that are arranged in parallel or inseries with respect to each other.
 25. The structure of claim 19 whereinthe tank circuit comprises an inductive element and a capacitive elementthat are arranged in parallel with respect to each other and wherein thetank circuit has an inductance between about 50 nanohenries and about250 nanohenries.
 26. The structure of claim 19 wherein the tank circuitcomprises an inductive element and a capacitive element that arearranged in parallel with respect to each other and wherein the tankcircuit has a capacitance between about 50 picofarads and about 2000picofarads.
 27. The structure of claim 19 wherein the tank circuitcomprises an inductive element and a capacitive element that arearranged in parallel with respect to each other and wherein the tankcircuit has a resonance frequency between about 10 megahertz and about20 megahertz.
 28. The structure of claim 19 wherein the tank circuit hasa resonance frequency f_(T) that is determined by the equality:f _(T)=ω_(T)/2π=1/{square root}{square root over (L_(T)C_(T))} whereinL_(T) is an inductance of the tank circuit; and C_(T) is a capacitanceof the tank circuit.
 29. The structure of claim 19, wherein the tankcircuit comprises one or more Josephson junctions.
 30. The structure ofclaim 29, the structure further comprising means for biasing the one ormore Josephson junctions of the tank circuit.
 31. The structure of claim19, the structure further comprising an amplifier connected across thetank circuit in such a manner that the amplifier can detect a change involtage across the tank circuit.
 32. The structure of claim 31, whereinthe amplifier comprises a high electron mobility field-effect transistor(HEMT) or a pseudomorphic high electron mobility field-effect transistor(PHEMT).
 33. The structure of claim 31, wherein the amplifier comprisesa multi-stage amplifier.
 34. The structure of claim 33 wherein themulti-stage amplifier comprises two, three, or four transistors
 35. Thestructure of claim 19 the structure further comprising a helium-3 pot ofa dilution refrigerator that is thermally coupled to all or a portion ofthe plurality of superconducting qubits.
 36. The structure of claim 19,the structure further comprising means for driving the tank circuit by adirect bias current I_(DC).
 37. The structure of claim 36, the structurefurther comprising means for driving the tank circuit by an alternatingcurrent I_(RF) of a frequency co close to the resonance frequency ω₀ ofthe tank circuit.
 38. The structure of claim 37 wherein the totalexternally applied magnetic flux to a superconducting qubit in theplurality of superconducting qubits, Φ_(E), isΦ_(E)=Φ_(DC)+Φ_(RF) wherein Φ_(RF) is an amount of applied magnetic fluxcontributed to the superconducting qubit by the alternating currentI_(RF); and Φ_(DC) is an amount of applied magnetic flux that isdetermined by the direct bias current I_(DC).
 39. The structure of claim38, the structure further comprising means for applying a magnetic fieldon the superconducting qubit, and whereinΦ_(DC)=Φ_(A) +f(t)Φ₀, wherein, Φ₀ is one flux quantum; f(t)Φ₀ isconstant or is slowly varying and is generated by the direct biascurrent I_(DC); andΦ_(A) =B _(A) ×L _(Q), wherein, B_(A) is a magnitude of the magneticfield applied on the superconducting qubit by the means for applying themagnetic field; and L_(Q) is an inductance of the superconducting qubit.40. The structure of claim 39, wherein f(t) has a value between 0 and 1inclusive.
 41. The structure of claim 39, wherein the means for applyinga magnetic field on the superconducting qubit comprises a bias line thatis magnetically coupled to said superconducting qubit.
 42. The structureof claim 39, wherein the means for applying a magnetic field on thesuperconducting qubit is an excitation device.
 43. The structure ofclaim 38, wherein Φ_(RF) has a magnitude between about 10⁻⁵Φ₀ and about10⁻¹Φ₀.
 44. The structure of claim 39, the structure further comprisingmeans for varying f(t), Φ_(A), and/or Φ_(RF).
 45. The structure of claim39, the structure further comprising means for varying Φ_(RF) inaccordance with a small amplitude fast function.
 46. The structure ofclaim 39, wherein the means for varying Φ_(RF) in accordance with asmall amplitude fast function is a microwave generator that is inelectrical communication with said tank circuit.
 47. The structure ofclaim 37, the structure further comprising an amplifier connected acrossthe tank circuit; and means for measuring a total impedance of the tankcircuit, expressed through the phase angle χ between driving currentI_(RF) and the tank voltage.
 48. The structure of claim 47, wherein themeans for measuring a total impedance of the tank circuit is anoscilloscope.